May 07, 2026
We introduce the geometric mean and the parallel sum of completely positive (CP) maps on von Neumann algebras, based on the Pusz–Woronowicz theory of positive sesquilinear forms. We provide a concrete characterization via a block matrix positivity condition and establish their fundamental properties, including the AM–GM–HM inequality with respect to the CP order.
In finite-dimensional settings, our construction is compatible with the Choi–Jamiołkowski correspondence, under which the geometric mean of CP maps corresponds to the Kubo–Ando geometric mean of their Choi matrices. This yields a natural operator-theoretic framework for interpolating quantum channels.
As an application, we obtain index-type inequalities for conditional expectations in subfactor theory.
Finally, we establish a Lebesgue-type decomposition of CP maps via a parallel sum construction, thereby providing a unified framework that simultaneously generalizes Ando’s decomposition of bounded positive operators and Kosaki’s decomposition of normal positive functionals on von Neumann algebras.
Completely positive (CP) maps play a central role in operator algebras and quantum information theory, serving as the natural morphisms between noncommutative spaces. Despite their fundamental importance, a systematic theory of operator means and Lebesgue-type decompositions for CP maps has remained relatively underdeveloped, in contrast to the well-established theories for bounded positive operators and normal positive functionals.
The aim of this paper is to develop a unified geometric and order-theoretic framework for CP maps that extends both operator mean theory in the sense of Kubo–Ando [1] and Lebesgue-type decomposition theory in the sense of Ando [2] and Kosaki [3]. Our approach is based on the Pusz–Woronowicz theory [4], [5] of positive sesquilinear forms and the spatial realization of CP maps via the Stinespring dilation [6].
The starting point of our work is the observation that the geometric mean of positive sesquilinear forms provides a natural mechanism for interpolating between noncommutative structures without relying on multiplicative operations. This idea, originally introduced by Pusz and Woronowicz, has proved highly effective in mathematical physics, particularly in the study of relative entropy and quantum information geometry, as developed by Uhlmann [7] and Kosaki [8]–[10]. It is also closely related to subsequent developments in operator mean theory and noncommutative integration, for instance in the work of Hiai and Kosaki [11]. In this paper, we show that this formalism extends naturally to CP maps and gives rise to a rich order-theoretic and geometric structure.
Given two CP maps \(\Phi\) and \(\Psi\) between von Neumann algebras, we define their geometric mean \(\Phi \# \Psi\) through the associated positive sesquilinear forms. We establish a concrete characterization of this mean via a block matrix positivity condition, which yields an operator-inequality formulation analogous to the Kubo–Ando theory. In particular, we introduce the arithmetic, geometric, and harmonic means of CP maps and prove the AM–GM–HM inequality \[\Phi \, ! \, \Psi \le_{\mathrm{cp}} \Phi \# \Psi \le_{\mathrm{cp}} \Phi \nabla \Psi.\] We further show that these means satisfy fundamental structural properties such as joint monotonicity, concavity, and the transformer inequality.
In finite-dimensional settings, our construction is compatible with the Choi–Jamiołkowski correspondence [12], [13]. Namely, the geometric mean of CP maps corresponds to the Kubo–Ando geometric mean of their Choi matrices. This provides a natural operator-theoretic framework for interpolating quantum channels and reveals a structural mechanism for extracting their common features. Related aspects of operator means for CP maps have recently been investigated by Frenkel, Mosonyi, Vrana, and Weiner [14] in the context of quantum hypothesis testing.
As an application to operator algebras, we study geometric means of conditional expectations. While the geometric mean of conditional expectations is not itself a conditional expectation in general, we show that it admits a natural bimodule structure over the intersection algebra. We also obtain index-type inequalities that connect our construction to subfactor theory [15], [16].
Beyond the theory of means, we develop a theory of connections for CP maps that parallels the Kubo–Ando theory for bounded positive operators. This shows that many fundamental order-theoretic properties extend to the noncommutative setting of CP maps.
A central result of this paper is a canonical Lebesgue-type decomposition for CP maps, constructed via a parallel sum limit. This provides a unified framework that simultaneously generalizes Ando’s decomposition of bounded positive operators and Kosaki’s decomposition of normal positive functionals on von Neumann algebras.
Our approach is based on Arveson’s Radon–Nikodym theorem [17], which allows us to realize CP maps in a common Stinespring space and to interpret absolute continuity and singularity in terms of support projections in the commutant. In this framework, the absolutely continuous part is identified with a shorted operator, in the sense of Krein and its subsequent development by Anderson and Trapp [18]–[20].
A key feature of our formulation is that it reduces analytic conditions—such as range inclusion, density, and closability—to simple geometric relations between closed subspaces. In particular, the seemingly different criteria appearing in the works of Ando and Kosaki are shown to be manifestations of a single underlying projection-theoretic structure.
Taken together, these results reveal a rich geometric and order structure on the space of CP maps, providing a unified perspective that connects operator means, noncommutative integration, and quantum information theory.
This section recalls the necessary parts of the Pusz–Woronowicz theory of positive sesquilinear forms [4], fixing notation for later use.
Let \(\mathcal{V}\) be a complex vector space. We denote by \(F(\mathcal{V})\) the set of all sesquilinear forms on \(\mathcal{V}\), i.e., the set of all functions \(\mathcal{V}\times\mathcal{V}\to\mathbb{C}\), which are linear in the first and conjugate-linear in the second variable. Note that \(F(\mathcal{V})\) naturally forms a complex vector space. A sesquilinear form \(\alpha\in F(\mathcal{V})\) is called positive if \(\alpha(x, x)\geq 0\) for all \(x\in\mathcal{V}\). We denote by \(F_+(\mathcal{V})\) the set of all positive sesquilinear forms on \(\mathcal{V}\). Note that \(F_+(\mathcal{V})\) is a convex cone in \(F(\mathcal{V})\). For \(\alpha, \beta\in F(\mathcal{V})\), we define a partial order \(\alpha\geq \beta\) by \(\alpha-\beta\in F_+(\mathcal{V})\).
Let \(\mathcal{H}\) be a complex Hilbert space. We denote by \(\mathbb{B}(\mathcal{H})\) the C\(^*\)-algebra of all bounded linear operators on \(\mathcal{H}\). Let \(h\) be a linear map from \(\mathcal{V}\) onto a dense subspace of \(\mathcal{H}\), and let \(A\in\mathbb{B}(\mathcal{H})\). We say that \(\alpha\in F(\mathcal{V})\) is represented by \((h, A)\) if \[\alpha(x, y)=\mathopen{\langle}Ah(x), h(y)\mathclose{\rangle}\] for \(x, y\in\mathcal{V}\). In this case, we write \(\alpha\sim(h, A)\). Note that \(\alpha\) is positive if and only if \(A\) is positive.
If \(\alpha\sim(h,A)\) and \(\beta\sim(h,B)\) with \(AB=BA\), then we say that these representations are compatible. By [4], any pair of positive sesquilinear forms admits a compatible representation.
Let \(f\) be a locally bounded Borel function on \(\mathbb{R}_+^2=\{(r,s)\in\mathbb{R}^2 \mid r\geq 0, s\geq 0\}\). Recall that \(f\) is homogeneous if \[f(\lambda r, \lambda s)=\lambda f(r, s)\] for \(r, s, \lambda\in\mathbb{R}_+\). The following theorem establishes the Pusz–Woronowicz functional calculus for positive sesquilinear forms.
Theorem 1 ([4]). Let \(f\) be a homogeneous locally bounded Borel function on \(\mathbb{R}_+^2\). For \(\alpha, \beta\in F_+(\mathcal{V})\), the sesquilinear form \[\gamma(x, y)=\mathopen{\langle}f(A,B)h(x), h(y)\mathclose{\rangle}\] is independent of the choice of compatible representations \(\alpha\sim(h,A)\) and \(\beta\sim(h, B)\).
We denote by \(f(\alpha, \beta)\) the resulting sesquilinear form \(\gamma\) in Theorem 1. Note that the functions \[f_G(r,s)=\sqrt{rs} \quad\text{and}\quad f_P(r,s)=\frac{rs}{r+s}\;(\text{with the convention 0/0=0})\] are such homogeneous functions. Therefore, for \(\alpha, \beta\in F_+(\mathcal{V})\), we define the geometric mean \(\alpha\#\beta\) and the parallel sum \(\alpha : \beta\) by \[\alpha\#\beta\mathrel{\vcenter{:}}= f_G(\alpha, \beta) \quad\text{and}\quad \alpha : \beta\mathrel{\vcenter{:}}= f_P(\alpha, \beta),\] respectively We also define the harmonic mean \(\alpha\, !\, \beta\) by \[\alpha\, !\, \beta\mathrel{\vcenter{:}}= 2(\alpha : \beta).\]
Let \(\alpha, \beta, \gamma\in F_+(\mathcal{V})\). We say that \(\gamma\) is dominated by \(\{\alpha, \beta\}\) if \[|\gamma(x,y)|^2\leq\alpha(x,x)\beta(y,y)\] for \(x,y\in\mathcal{V}\).
Finally, we conclude this section with the following characterizations of the geometric mean and the parallel sum:
Theorem 2 ([4]). Let \(\alpha, \beta\in F_+(\mathcal{V})\). Then \[\alpha\#\beta = \max\{ \gamma\in F_+(\mathcal{V}) \mid \text{\gamma is dominated by \{\alpha, \beta\}} \}.\]
Lemma 1 ([4]). Let \(\alpha, \beta\in F_+(\mathcal{V})\). Then \[(\alpha : \beta)(z, z)= \inf\{ \alpha(x,x)+\beta(y,y)\mid z=x+y, x,y\in\mathcal{V} \} \quad\text{for}\;z\in\mathcal{V}.\]
Given \(\gamma_{ij}\in F(\mathcal{V})\) for \(1\leq i,j\leq n\), define the matrix form \(\Gamma=[\gamma_{ij}]\) on \(\mathcal{V}^n\) by \[\Gamma(x,y)=\sum_{i,j}\gamma_{ij}(x_j,y_i)\] for \(x=(x_1,\dots,x_n), y=(y_1,\dots,y_n)\in\mathcal{V}^n\). Note that \(\Gamma(x, x)\geq 0\) if and only if the scalar matrix \([\gamma_{ij}(x_j,x_i)]_{1\leq i,j\leq n}\) is positive. Then we can obtain the block matrix characterization of the geometric mean of positive sesquilinear forms.
Proposition 2. Let \(\alpha, \beta\in F_+(\mathcal{V})\). Then \[\alpha\# \beta = \max \left\{ \gamma\in F_+(\mathcal{V}) \;\middle|\; \begin{bmatrix} \alpha & \gamma \\ \gamma & \beta \\ \end{bmatrix}\geq 0 \right\}.\]
Proof. Let \(\gamma\in F_+(\mathcal{V})\). The proposition follows from Theorem 2 and the following fact: \[\begin{bmatrix} \alpha & \gamma \\ \gamma & \beta \\ \end{bmatrix}\geq 0 \quad \text{if and only if} \quad |\gamma(x,y)|^2\leq\alpha(x, x)\beta(y, y) \quad\text{for}\;x,y\in\mathcal{V}.\] ◻
Similarly, one can derive block matrix characterizations of the parallel sum and the harmonic mean from Lemma 1, using the variational formula for the parallel sum and a polarization argument.
Proposition 3. Let \(\alpha, \beta\in F_+(\mathcal{V})\). Then
\({\displaystyle \alpha : \beta=\max \left\{ \gamma\in F_+(\mathcal{V}) \;\middle|\; \begin{bmatrix} \alpha & \alpha \\ \alpha & \alpha+\beta \\ \end{bmatrix} \geq \begin{bmatrix} \gamma & 0 \\ 0 & 0 \\ \end{bmatrix} \right\}, }\)
\({\displaystyle \alpha\, !\, \beta=\max \left\{ \gamma\in F_+(\mathcal{V}) \;\middle|\; \begin{bmatrix} 2\alpha & 0 \\ 0& 2\beta \\ \end{bmatrix} \geq \begin{bmatrix} \gamma & \gamma \\ \gamma & \gamma \\ \end{bmatrix} \right\}.}\)
Let \(\mathcal{M}\) and \(\mathcal{N}\) be von Neumann algebras. To apply the Pusz–Woronowicz theory to completely positive (CP) maps from \(\mathcal{M}\) to \(\mathcal{N}\), we use the standard form of the target algebra \(\mathcal{N}\).
We review the theory of bimodules over von Neumann algebras. For further details and background, see [21]–[23].
Let \(L^2(\mathcal{N})\) be the standard form of \(\mathcal{N}\), and \(J\) be the modular conjugation. Note that \(L^2(\mathcal{N})\) has the \(\mathcal{N}\)-\(\mathcal{N}\) bimodule structure with left and right actions given by \[x\xi y\mathrel{\vcenter{:}}= xJ y^* J\xi\] for \(x, y\in\mathcal{N}\) and \(\xi\in L^2(\mathcal{N})\). Let us consider the \(n\times n\) matrix algebra \(M_n(\mathbb{C})\) with the canonical normalized trace \(\mathop{\mathrm{tr}}\). If we define the inner product on \(M_n(\mathbb{C})\) by \[\mathopen{\langle}x, y\mathclose{\rangle}\mathrel{\vcenter{:}}=\mathop{\mathrm{tr}}(y^*x)\] for \(x, y\in M_n(\mathbb{C})\), then \(M_n(\mathbb{C})\) can be regarded as a Hilbert space. Moreover, \(M_n(\mathbb{C})\) acts on itself from the left such that the modular conjugation is the canonical involution \(J_{\mathop{\mathrm{tr}}}\colon x\to x^*\). We also denote by \(J^{(n)}\) the modular conjugation on the standard form of \(M_n(\mathcal{N})=\mathcal{N}\otimes M_n(\mathbb{C})\).
We denote by \(\mathrm{CP}(\mathcal{M}, \mathcal{N})\) the set of all CP maps from \(\mathcal{M}\) to \(\mathcal{N}\). For \(\Phi,\Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), we define a partial order \(\Phi\geq_{\mathrm{cp}}\Psi\) if \(\Phi-\Psi\) is completely positive.
For \(\Phi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), we define a positive sesquilinear form \(s_\Phi\) on the algebraic tensor product \(\mathcal{M}\odot L^2(\mathcal{N})\) by \[s_\Phi(x \otimes \xi, y \otimes \eta) \mathrel{\vcenter{:}}= \langle \Phi(y^* x) \xi, \eta \rangle\] for \(x, y\in\mathcal{M}\) and \(\xi, \eta\in L^2(\mathcal{N})\). Note that, for \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), \(\Phi\leq_{\mathrm{cp}}\Psi\) if and only if \(s_\Phi\leq s_\Psi\). Via separation and completion, we construct the Hilbert space \(\mathcal{H}_\Phi\). The left action of \(\mathcal{M}\) and the right action of \(\mathcal{N}\) are defined by \[a(x\otimes_\Phi \xi)b\mathrel{\vcenter{:}}= (ax)\otimes_\Phi (\xi b)\] for \(a, x\in \mathcal{M}\), \(b\in\mathcal{N}\) and \(\xi\in L^2(\mathcal{N})\).
We now give a short proof of [4] in our setting. For \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), we set \(\alpha=s_\Phi\) and \(\beta=s_\Psi\).
Put \(\gamma=\alpha+\beta\). For \(b\in \mathcal{N}\), we have \[\label{eq:right} \gamma(x\otimes (\xi b), y\otimes \eta) = \gamma(x\otimes \xi, y\otimes (\eta b^*)).\tag{1}\] Via separation and completion of \(\mathcal{M}\odot L^2(\mathcal{N})\), we construct a Hilbert space \(\mathcal{H}_\gamma\). Denote by \(h\) the canonical map from \(\mathcal{M}\odot L^2(\mathcal{N})\) to \(\mathcal{H}_\gamma\). Then there exist bounded positive operators \(A, B\) on \(\mathcal{H}_\gamma\) such that \[\alpha(x\otimes \xi, y\otimes \eta) = \mathopen{\langle} Ah(x\otimes \xi), h(y\otimes \eta) \mathclose{\rangle}_\gamma\] and \[\beta(x\otimes \xi, y\otimes \eta) = \mathopen{\langle} Bh(x\otimes \xi), h(y\otimes \eta) \mathclose{\rangle}_\gamma.\] Note that \(A+B=I\). In particular, \(A\) and \(B\) commute. For \(b\in \mathcal{N}\), We can define a bounded operator \(\rho(b)\) on \(\mathcal{H}_\gamma\) by \[\rho(b)h(x\otimes \xi)=h(x\otimes (\xi b)),\] for \(x\in\mathcal{M}\) and \(\xi\in L^2(\mathcal{N})\), because \[\begin{align} \mathopen{\langle}h(x\otimes (\xi b)), h(x\otimes (\xi b))\mathclose{\rangle}_\gamma &= \alpha(x\otimes (\xi b), x\otimes (\xi b)) + \beta(x\otimes (\xi b), x\otimes (\xi b)) \\ &\leq \|b\|^2\Big(\alpha(x\otimes \xi, x\otimes \xi) + \beta(x\otimes \xi, x\otimes \xi)\Big) \\ &= \|b\|^2\mathopen{\langle}h(x\otimes \xi), h(x\otimes \xi)\mathclose{\rangle}_\gamma. \end{align}\] By using (1 ), we obtain \(\rho(b)^*=\rho(b^*)\). Moreover, \[\begin{align} \mathopen{\langle}A\rho(b)h(x\otimes \xi), h(y\otimes \eta)\mathclose{\rangle}_\gamma &= \mathopen{\langle}Ah(x\otimes (\xi b)), h(y\otimes \eta)\mathclose{\rangle}_\gamma \\ &= \alpha(x\otimes (\xi b), y\otimes \eta) \\ &= \alpha(x\otimes \xi, y\otimes (\eta b^*)) \\ &= \mathopen{\langle}Ah(x\otimes \xi), \rho(b^*)h(y\otimes \eta)\mathclose{\rangle}_\gamma \\ &= \mathopen{\langle}\rho(b)Ah(x\otimes \xi), h(y\otimes \eta)\mathclose{\rangle}_\gamma. \end{align}\] Thus \(\rho(b)A=A\rho(b)\). Similarly, we obtain \(\rho(b)B=B\rho(b)\). Hence, for any homogeneous locally bounded Borel function \(f\) on \(\mathbb{R}_+^2\), we have \[f(A, B)\rho(b)=\rho(b)f(A, B).\] Therefore, for \(b\in\mathcal{N}\), \[\label{eq:right-action} f(s_\Phi, s_\Psi)(x\otimes (\xi b), y\otimes \eta)= f(s_\Phi, s_\Psi)(x\otimes \xi, y\otimes (\eta b^*)).\tag{2}\] Similarly, we also have, for \(a\in\mathcal{M}\), \[\label{eq:left-action} f(s_\Phi, s_\Psi)((ax)\otimes \xi, y\otimes \eta)= f(s_\Phi, s_\Psi)(x\otimes \xi, (a^*y)\otimes \eta).\tag{3}\] These equations play a crucial role in recovering a cp map from the resulting sesquilinear form.
Let \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\). By Pusz–Woronowicz theory, we obtain the geometric mean \(\gamma=s_\Phi\# s_\Psi\) on \(\mathcal{M}\odot L^2(\mathcal{N})\). Since the positive sesquilinear form \(\gamma\) is dominated by \(\{s_\Phi, s_\Psi\}\), for \(x, y\in \mathcal{M}\) and \(\xi, \eta\in L^2(\mathcal{N})\), we have \[\begin{align} |\gamma(x\otimes \xi, y\otimes \eta)|^2 &\leq s_\Phi(x\otimes \xi, x\otimes \xi)s_\Psi(y\otimes \eta, y\otimes \eta) \\ &= \mathopen{\langle} \Phi(x^*x)\xi, \xi \mathclose{\rangle} \mathopen{\langle} \Psi(y^*y)\eta, \eta \mathclose{\rangle}. \end{align}\] We define a Hilbert space \(\mathcal{H}_\gamma\) via separation and completion of \(\mathcal{M}\odot L^2(\mathcal{N})\) with respect to \(\gamma\). We denote by \(h\colon \mathcal{M}\odot L^2(\mathcal{N})\to\mathcal{H}_\gamma\) the canonical map.
We first claim that the Hilbert space \(\mathcal{H}_\gamma\) has the left action of \(\mathcal{M}\) and the right action of \(\mathcal{N}\). For a fixed \(a\in\mathcal{M}\), we define the positive sesquilinear form \(\gamma^a\) on \(\mathcal{M}\odot L^2(\mathcal{N})\) by \[\gamma^a(x\otimes \xi, y\otimes \eta) \mathrel{\vcenter{:}}= \gamma((ax)\otimes \xi, (ay)\otimes \eta)\] for \(x, y\in\mathcal{M}\) and \(\xi, \eta\in L^2(\mathcal{N})\). Since \[\begin{align} |\gamma^a(x\otimes \xi, y\otimes \eta)|^2 &\leq \mathopen{\langle} \Phi(x^*a^*ax)\xi, \xi \mathclose{\rangle} \mathopen{\langle} \Psi(y^*a^*ay)\eta, \eta) \mathclose{\rangle} \\ &\leq \|a\|^4s_\Phi(x\otimes \xi, x\otimes \xi) s_\Psi(y\otimes \eta, y\otimes \eta), \end{align}\] the positive sesquilinear form \(\gamma^a\) is dominated by \(\{\|a\|^2s_\Phi, \|a\|^2s_\Psi\}\). Hence, \[\gamma^a\leq(\|a\|^2s_\Phi)\#(\|a\|^2s_\Psi) =\|a\|^2(s_\Phi\# s_\Psi)=\|a\|^2\gamma.\] Therefore, we can define the left action of \(\mathcal{M}\) on \(\mathcal{H}_\gamma\) by \[a(x\otimes \xi)\mathrel{\vcenter{:}}= (ax)\otimes \xi\] for \(a, x\in\mathcal{M}\) and \(\xi\in L^2(\mathcal{N})\). Similarly, we can also define the right action of \(\mathcal{N}\) on \(\mathcal{H}_\gamma\) by \[(x\otimes \xi)b\mathrel{\vcenter{:}}= x\otimes (\xi b)\] for \(x\in\mathcal{M}\), \(b\in \mathcal{N}\) and \(\xi\in L^2(\mathcal{N})\). By (2 ), for \(b\in\mathcal{N}\), we have \[\gamma(x\otimes (\xi b), y\otimes \eta)= \gamma(x\otimes \xi, y\otimes (\eta b^*)).\]
Next we claim that an operator \[V\colon L^2(\mathcal{N})\to\mathcal{H}_\gamma, \quad \xi\mapsto h(1\otimes \xi)\] is bounded. Indeed, \[\begin{align} |\gamma(1\otimes \xi, 1\otimes \xi)|^2 &\leq \mathopen{\langle} \Phi(1)\xi, \xi \mathclose{\rangle} \mathopen{\langle} \Psi(1)\xi, \xi \mathclose{\rangle} \\ &\leq \|\Phi\|\|\Psi\|\|\xi\|^4. \end{align}\] Hence, we obtain the bounded operator \(V\). Then we define the CP map \(\Theta\) on \(\mathcal{M}\) by \[\Theta(x)=V^*xV\] for \(x\in\mathcal{M}\).
We finally show that \(\Theta(x)\in\mathcal{N}\) for \(x\in \mathcal{M}\). Since, for \(b\in\mathcal{N}\), \[\begin{align} \mathopen{\langle} \Theta(x)Jb^*J\xi, \eta \mathclose{\rangle} &= \mathopen{\langle} xV(\xi b), V\eta \mathclose{\rangle} \\ &= \gamma(x\otimes (\xi b), 1\otimes \eta) \\ &= \gamma(x\otimes \xi, 1\otimes (\eta b^*)) \\ &= \mathopen{\langle} xV\xi, V(\eta b^*) \mathclose{\rangle} \\ &= \mathopen{\langle} \Theta(x)\xi, \eta b^* \mathclose{\rangle} \\ &= \mathopen{\langle} Jb^*J\Theta(x)\xi, \eta \mathclose{\rangle}, \end{align}\] we obtain \(\Theta(x)\in(\mathcal{N}')'=\mathcal{N}\). By (3 ), we also have \[\mathopen{\langle} \Theta(y^*x)\xi, \eta \mathclose{\rangle} = (s_\Phi \# s_\Psi)(x\otimes \xi, y\otimes \eta).\]
Definition 1. For \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), the geometric mean \(\Phi\#\Psi\) is defined by \[\Phi\#\Psi\mathrel{\vcenter{:}}= \Theta\in\mathrm{CP}(\mathcal{M}, \mathcal{N}).\]
By the definition, one can also see that if \(\Phi\) and \(\Psi\) are norml, then so is \(\Phi\#\Psi\). The following can be easily verified.
Proposition 4. Let \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then the following conditions hold:
\(\Phi\#\Psi=\Psi\#\Phi\),
\(\Phi\#\Phi=\Phi\),
\(\|\Phi\#\Psi\|\leq\|\Phi\|^{1/2}\|\Psi\|^{1/2}\).
Remark 5. Our construction of the geometric mean for CP maps avoids the approximation procedures typically required in the Kubo-Ando framework such as considering \(\lim_{\varepsilon\downarrow 0}(A + \varepsilon I)\) for non-invertible operators. By defining the geometric mean through the Pusz–Woronowicz theory of sesquilinear forms on the standard representation, we can directly handle general CP maps.
Remark 6. Let \(\mathcal{M}=\mathbb{C}\) and \(\mathcal{N}=\mathbb{B}(\mathcal{H})\), where \(\mathcal{H}\) is an infinite dimensional Hilbert space. For each \(A\in\mathbb{B}(\mathcal{H})^+\), we define a CP map \(\Phi_A\colon \mathbb{C}\to\mathbb{B}(\mathcal{H})\) by \[\Phi_A(z)\mathrel{\vcenter{:}}= z A\] for \(z\in\mathbb{C}\). Then the corresponding sesquilinear form \(s_{\Phi_A}\) on \(\mathbb{C}\otimes\mathcal{H}\cong\mathcal{H}\) is given by \[s_{\Phi_A}(z\otimes\xi, w\otimes\eta) =z\bar w\mathopen{\langle} A\xi, \eta \mathclose{\rangle}.\] For any \(A, B\in\mathbb{B}(\mathcal{H})^+\), the geometric mean of the sesquilinear forms \(s_{\Phi_A}\) and \(s_{\Phi_B}\) satisfies \[s_{\Phi_A}\#s_{\Phi_B}(z\otimes\xi, w\otimes\eta) = z\bar w \mathopen{\langle} (A\# B)\xi, \eta \mathclose{\rangle},\] where \(A\# B\) denotes the geometric mean of bounded positive operators. Consequently, we obtain \[\Phi_A\#\Phi_B=\Phi_{A\# B}.\] This shows that our definition of the geometric mean for CP maps is a natural extension of the classical operator version.
Next, we give a block matrix characterization of the geometric mean \(\Phi\#\Psi\). Let \(\Phi, \Psi, \Theta\in\mathrm{CP}(\mathcal{M},\mathcal{N})\). We define the linear maps \(\Pi\colon\mathcal{M}\to M_2(\mathcal{N})\) and \(\tilde{\Pi}\colon M_2(\mathcal{N})\to M_2(\mathcal{N})\) by \[\Pi(x)\mathrel{\vcenter{:}}= \begin{bmatrix} \Phi(x) & \Theta(x) \\ \Theta(x) & \Psi(x) \\ \end{bmatrix}\quad \text{and}\quad \tilde{\Pi} \begin{bmatrix} x_{11} & x_{12} \\ x_{21} & x_{22} \\ \end{bmatrix}\mathrel{\vcenter{:}}= \begin{bmatrix} \Phi(x_{11}) & \Theta(x_{12}) \\ \Theta(x_{21}) & \Phi(x_{22}) \\ \end{bmatrix},\] respectively. By [24], we have \(\Pi\) is CP if and only if \(\tilde{\Pi}\) is CP.
Proposition 7. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then \[\Phi \# \Psi = \max \left\{ \Theta \in \mathrm{CP}(\mathcal{M}, \mathcal{N}) \,\middle|\, \begin{bmatrix} \Phi & \Theta \\ \Theta & \Psi \\ \end{bmatrix} \geq_{\mathrm{cp}} 0 \right\}.\]
Proof. Suppose that \(\Pi\geq_{\mathrm{cp}}0\). Then \(\tilde{\Pi}\geq_{\mathrm{cp}}0\). Recall that \(\tilde{\Pi}\) is CP if and only if \(s_{\tilde{\Pi}}\) is positive. Since \(s_{\tilde{\Pi}}\) is positive, for \(\zeta_1, \zeta_2\in \mathcal{M}\odot L^2(\mathcal{N})\), we have \[|s_\Theta(\zeta_1, \zeta_2)|^2 \leq s_\Phi(\zeta_1, \zeta_1) s_\Psi(\zeta_2, \zeta_2).\] In fact, for simple tensors \(\zeta_1=x_1\otimes \xi_1\) and \(\zeta_2=x_2\otimes \xi_2\), if \[\boldsymbol{x}= \begin{bmatrix} x_1 & x_2 \\ 0 & 0 \\ \end{bmatrix}\quad\text{and}\quad \boldsymbol{\xi} = \begin{bmatrix} \xi_1 & 0 \\ \xi_2 & 0 \\ \end{bmatrix},\] then \[\begin{align} 0\leq s_{\tilde{\Pi}}(\boldsymbol{x} \otimes \boldsymbol{\xi}, \boldsymbol{x} \otimes \boldsymbol{\xi}) &= \mathopen{\langle} \tilde{\Pi}(\boldsymbol{x}^*\boldsymbol{x})\boldsymbol{\xi}, \boldsymbol{\xi} \mathclose{\rangle} \\ &= \mathopen{\langle} \begin{bmatrix} \Phi(x_1^*x_1) & \Theta(x_1^*x_2) \\ \Theta(x_2^*x_1) & \Psi(x_2^*x_2) \\ \end{bmatrix} \begin{bmatrix} \xi_1 & 0 \\ \xi_2 & 0 \\ \end{bmatrix}, \begin{bmatrix} \xi_1 & 0 \\ \xi_2 & 0 \\ \end{bmatrix} \mathclose{\rangle} \\ &= \mathopen{\langle} \Phi(x_1^*x_1)\xi_1, \xi_1 \mathclose{\rangle} + \mathopen{\langle} \Theta(x_1^*x_2)\xi_2, \xi_1 \mathclose{\rangle} \\ &\quad\quad + \mathopen{\langle} \Theta(x_2^*x_1)\xi_1, \xi_2 \mathclose{\rangle} + \mathopen{\langle} \Psi(x_2^*x_2)\xi_2, \xi_2 \mathclose{\rangle}. \end{align}\] Therefore, \[|s_\Theta(x_1 \otimes \xi_1, x_2 \otimes \xi_2)|^2 \leq s_\Phi(x_1 \otimes \xi_1, x_1 \otimes \xi_1) s_\Psi(x_2 \otimes \xi_2, x_2 \otimes \xi_2).\] Hence, by the maximality of the geometric mean, we have \[s_\Theta \leq s_{\Phi \# \Psi},\] which implies \[\Theta \leq_{\mathrm{cp}} \Phi \# \Psi.\]
Now we put \(\Theta = \Phi \# \Psi\). We next show that \(s_\Pi\) is positive. Let \(x\in \mathcal{M}\) and \[\boldsymbol{\xi} = \begin{bmatrix} \eta_1 & \eta_2 \\ \zeta_1 & \zeta_2 \end{bmatrix}\in M_2(L^2(\mathcal{N})).\] By linearity and polarization, it suffices to check that \[s_\Pi(x \otimes \boldsymbol{\xi}, x \otimes \boldsymbol{\xi}) \geq 0.\] Note that \[\begin{align} s_\Pi(x \otimes \boldsymbol{\xi}, x \otimes \boldsymbol{\xi}) &= \sum_{i=1,2}\Big\{\mathopen{\langle}\Phi(x^*x)\eta_i, \eta_i\mathclose{\rangle} + \mathopen{\langle}\Theta(x^*x)\zeta_i, \eta_i\mathclose{\rangle} \\ &\qquad+ \mathopen{\langle}\Theta(x^*x)\eta_i, \zeta_i\mathclose{\rangle} + \mathopen{\langle}\Psi(x^*x)\zeta_i, \zeta_i\mathclose{\rangle} \Big\}. \end{align}\] Since \(s_{\Phi \# \Psi} = s_\Phi \# s_\Psi\), by the definition of the geometric mean, we have \[\mathopen{\langle}\Phi(x^*x)\eta_i, \eta_i\mathclose{\rangle} + \mathopen{\langle}\Theta(x^*x)\zeta_i, \eta_i\mathclose{\rangle} + \mathopen{\langle}\Theta(x^*x)\eta_i, \zeta_i\mathclose{\rangle} + \mathopen{\langle}\Psi(x^*x)\zeta_i, \zeta_i\mathclose{\rangle} \geq 0\] for \(i=1,2\). Therefore, \(s_\Pi\) is positive, which means \(\Pi\) is CP. Thus the proposition follows. ◻
Theorem 3. Let \(\Phi, \Psi, \Phi_i, \Psi_i \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) for \(i=1,2\). The following properties hold:
\(\Phi_1 \leq_{\mathrm{cp}} \Phi_2\) and \(\Psi_1 \leq_{\mathrm{cp}} \Psi_2\) imply \((\Phi_1 \# \Psi_1) \leq_{\mathrm{cp}} (\Phi_2 \# \Psi_2)\),
For any von Neumann algebra \(\mathcal{P}\) and \(\Xi \in \mathrm{CP}(\mathcal{N}, \mathcal{P})\), \[\Xi \circ (\Phi \# \Psi) \leq_{\mathrm{cp}} (\Xi \circ \Phi) \# (\Xi \circ \Psi),\]
For any von Neumann algebra \(\mathcal{L}\) and \(\Xi \in \mathrm{CP}(\mathcal{L}, \mathcal{M})\), \[(\Phi \# \Psi) \circ \Xi \leq_{\mathrm{cp}} (\Phi \circ \Xi) \# (\Psi \circ \Xi),\]
\(\Phi_n, \Psi_n \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\), \(\Phi_n \downarrow \Phi\) and \(\Psi_n \downarrow \Psi\) imply \(\Phi_n \# \Psi_n \downarrow \Phi \# \Psi\), where \(\Phi_n \downarrow \Phi\) means that \(\Phi_1 \geq_{\mathrm{cp}} \Phi_2 \geq_{\mathrm{cp}} \cdots\) and \(\Phi_n(x) \to \Phi(x)\) in the ultraweak operator topology for each \(x \in \mathcal{M}^+\).
Proof. (1) Let \(\Theta = \Phi_1 \# \Psi_1\). By Proposition 7, we have \[\begin{bmatrix} \Phi_1 & \Theta \\ \Theta & \Psi_1 \\ \end{bmatrix} \geq_{\mathrm{cp}} 0.\] Since \(\Phi_1 \leq_{\mathrm{cp}} \Phi_2\) and \(\Psi_1 \leq_{\mathrm{cp}} \Psi_2\), we obtain \[\begin{bmatrix} \Phi_2 & \Theta \\ \Theta & \Psi_2 \\ \end{bmatrix} = \begin{bmatrix} \Phi_1 & \Theta \\ \Theta & \Psi_1 \\ \end{bmatrix} + \begin{bmatrix} \Phi_2 - \Phi_1 & 0 \\ 0 & \Psi_2 - \Psi_1 \\ \end{bmatrix} \geq_{\mathrm{cp}} 0.\] By Proposition 7, this implies that \(\Theta \leq_{\mathrm{cp}} \Phi_2 \# \Psi_2\), which is the desired conclusion.
(2) Let \(\Xi \in \mathrm{CP}(\mathcal{N}, \mathcal{P})\). By applying the CP map \(\Xi^{(2)} = \mathrm{id}_{M_2(\mathbb{C})} \otimes \Xi\) to \[\begin{bmatrix} \Phi & \Phi \# \Psi \\ \Phi \# \Psi & \Psi \\ \end{bmatrix}\geq_{\mathrm{cp}} 0,\] we obtain \[\begin{bmatrix} \Xi \circ \Phi & \Xi \circ (\Phi \# \Psi) \\ \Xi \circ (\Phi \# \Psi) & \Xi \circ \Psi \\ \end{bmatrix} = \Xi^{(2)} \circ \begin{bmatrix} \Phi & \Phi \# \Psi \\ \Phi \# \Psi & \Psi \\ \end{bmatrix} \geq_{\mathrm{cp}} 0,\] which implies \[\Xi \circ (\Phi \# \Psi) \leq_{\mathrm{cp}} (\Xi \circ \Phi) \# (\Xi \circ \Psi).\]
(3) Let \(\Xi \in \mathrm{CP}(\mathcal{L}, \mathcal{M})\). Since \[\begin{bmatrix} \Phi \circ \Xi & (\Phi \# \Psi) \circ \Xi \\ (\Phi \# \Psi) \circ \Xi & \Psi \circ \Xi \\ \end{bmatrix} = \begin{bmatrix} \Phi & \Phi \# \Psi \\ \Phi \# \Psi & \Psi \\ \end{bmatrix} \circ \Xi \geq_{\mathrm{cp}} 0,\] we obtain \[(\Phi \# \Psi) \circ \Xi \leq_{\mathrm{cp}} (\Phi \circ \Xi) \# (\Psi \circ \Xi).\]
(4) Since \(\Phi_n \geq_{\mathrm{cp}} \Phi_{n+1}\) and \(\Psi_n \geq_{\mathrm{cp}} \Psi_{n+1}\), property (1) implies that \[\Phi_n \# \Psi_n \geq_{\mathrm{cp}} \Phi_{n+1} \# \Psi_{n+1}.\] Furthermore, since \(\Phi_n \geq_{\mathrm{cp}} \Phi\) and \(\Psi_n \geq_{\mathrm{cp}} \Psi\), we also have \[\Phi_n \# \Psi_n \geq_{\mathrm{cp}} \Phi \# \Psi \quad \text{for all } n.\] Thus, for each \(x \in \mathcal{M}^+\), the sequence \(\{(\Phi_n \# \Psi_n)(x)\}_{n=1}^\infty\) is monotonically decreasing in \(\mathcal{N}^+\), and is bounded below by \((\Phi \# \Psi)(x)\). Therefore, it converges in the ultraweak operator topology. Since every element in \(\mathcal{M}\) is a linear combination of four positive elements, we can define a linear map \(\Theta \colon \mathcal{M}\to \mathcal{N}\) by the point-weak limit \[\Theta(x) \mathrel{\vcenter{:}}= \lim_{n \to \infty} (\Phi_n \# \Psi_n)(x)\] for \(x \in \mathcal{M}\). Because \(\mathrm{CP}(\mathcal{M},\mathcal{N})\) is closed under the point-ultraweak operator topology, \(\Theta\) is CP. By construction, we have \[\Theta \geq_{\mathrm{cp}} \Phi \# \Psi.\] To show the reverse inequality, we use the block matrix characterization. For each \(n\), we have \[\begin{bmatrix} \Phi_n & \Phi_n \# \Psi_n \\ \Phi_n \# \Psi_n & \Psi_n \\ \end{bmatrix} \geq_{\mathrm{cp}} 0.\] Taking the limit as \(n \to \infty\) in the point-ultraweak operator topology, we obtain \[\begin{bmatrix} \Phi & \Theta \\ \Theta & \Psi \\ \end{bmatrix} \geq_{\mathrm{cp}} 0.\] By Proposition 7, this implies that \(\Theta \leq_{\mathrm{cp}} \Phi \# \Psi\). Therefore, we conclude that \(\Theta = \Phi \# \Psi\), which means \(\Phi_n \# \Psi_n \downarrow \Phi \# \Psi\). ◻
Corollary 1. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Suppose that \(\alpha \in \mathrm{Aut}(\mathcal{N})\) and \(\beta \in \mathrm{Aut}(\mathcal{M})\). Then \[\alpha \circ (\Phi \# \Psi) \circ \beta = (\alpha \circ \Phi \circ \beta) \# (\alpha \circ \Psi \circ \beta).\]
Proof. By Theorem 3, we immediately have \[\alpha \circ (\Phi \# \Psi) \circ \beta \leq_{\mathrm{cp}} (\alpha \circ \Phi \circ \beta) \# (\alpha \circ \Psi \circ \beta).\] Applying Theorem 3 again, we obtain \[\begin{align} &\alpha^{-1} \circ \left( (\alpha \circ \Phi \circ \beta) \# (\alpha \circ \Psi \circ \beta) \right) \circ \beta^{-1} \\ &\quad\leq_{\mathrm{cp}} (\alpha^{-1} \circ \alpha \circ \Phi \circ \beta \circ \beta^{-1}) \# (\alpha^{-1} \circ \alpha \circ \Psi \circ \beta \circ \beta^{-1}) \\ &\quad= \Phi \# \Psi. \end{align}\] Applying \(\alpha\) from the left and \(\beta\) from the right to both sides of this inequality, we get \[(\alpha \circ \Phi \circ \beta) \# (\alpha \circ \Psi \circ \beta) \leq_{\mathrm{cp}} \alpha \circ (\Phi \# \Psi) \circ \beta.\] Combining these two inequalities yields the desired equality. ◻
Proposition 8. Let \(\Phi_i, \Psi_i\in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) for \(i=1,2\). Then \[(\Phi_1\#\Psi_1)+(\Phi_2\#\Psi_2)\leq(\Phi_1+\Phi_2)\#(\Psi_1+\Psi_2).\]
Proof. Suppose that \(\Theta_i\in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) satisfies \[\begin{bmatrix} \Phi_i & \Theta_i \\ \Theta_i & \Psi_i \\ \end{bmatrix}\geq_{\mathrm{cp}} 0\] for \(i=1,2\). Then \[\begin{bmatrix} \Phi_1+\Phi_2& \Theta_1+\Theta_2\\ \Theta_1+\Theta_2 & \Psi_1+\Psi_2\\ \end{bmatrix}\geq_{\mathrm{cp}} 0.\] By Proposition 7, \[\Theta_1+\Theta_2\leq_{\mathrm{cp}}(\Phi_1+\Phi_2)\#(\Psi_1+\Psi_2).\] ◻
We next consider the parallel sum for CP maps. The following is essentially the same argument as for the geometric mean, but we provide a brief outline for the reader.
Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). By the Pusz–Woronowicz theory, we obtain the parallel sum \(\gamma = s_\Phi : s_\Psi\) on \(\mathcal{M}\odot L^2(\mathcal{N})\). We define a Hilbert space \(\mathcal{H}_\gamma\) via separation and completion of \(\mathcal{M}\odot L^2(\mathcal{N})\) with respect to \(\gamma\). We denote by \(h\colon \mathcal{M}\odot L^2(\mathcal{N})\to\mathcal{H}_\gamma\) the canonical map.
We first claim that the Hilbert space \(\mathcal{H}_\gamma\) has the left action of \(\mathcal{M}\) and the right action of \(\mathcal{N}\). For a fixed \(a\in\mathcal{M}\), we define the sesquilinear form \(\gamma^a\) on \(\mathcal{M}\odot L^2(\mathcal{N})\) by \[\gamma^a(x\otimes \xi, y\otimes \eta) \mathrel{\vcenter{:}}= \gamma((ax)\otimes \xi, (ay)\otimes \eta)\] for \(x, y\in\mathcal{M}\) and \(\xi, \eta\in L^2(\mathcal{N})\). Suppose that \(\zeta, \zeta_1, \zeta_2\in\mathcal{M}\odot L^2(\mathcal{N})\) such that \(\zeta=\zeta_1+\zeta_2\). Then by Lemma 1, \[\gamma(\zeta, \zeta)\leq s_\Phi(\zeta_1, \zeta_1)+s_\Psi(\zeta_2, \zeta_2).\] Since \(a\zeta=a \zeta_1+a\zeta_2\), we have \[\begin{align} \gamma^a(\zeta, \zeta) &= \gamma(a\zeta, a\zeta) \\ &\leq s_\Phi(a\zeta_1, a\zeta_1)+ s_\Psi(a\zeta_2, a\zeta_2) \\ &\leq \|a\|^2s_\Phi(\zeta_1, \zeta_1) +\|a\|^2s_\Psi(\zeta_2, \zeta_2). \end{align}\] By taking the infimum over \(\zeta=\zeta_1+\zeta_2\), we obtain \[\gamma^a\leq\|a\|^2(s_\Phi : s_\Psi).\] Therefore, we can define the left action of \(\mathcal{M}\) on \(\mathcal{H}_\gamma\) by \[a(x\otimes \xi)\mathrel{\vcenter{:}}= (ax)\otimes \xi\] for \(x\in\mathcal{M}\) and \(\xi\in L^2(\mathcal{N})\). Similarly, we can also define the right action of \(\mathcal{N}\) on \(\mathcal{H}_\gamma\) by \[(x\otimes \xi)b\mathrel{\vcenter{:}}= x\otimes (\xi b)\] for \(x\in\mathcal{M}\) and \(\xi\in L^2(\mathcal{N})\).
Next we claim that an operator \[V\colon L^2(\mathcal{N})\to\mathcal{H}_\gamma, \quad \xi\mapsto h(1\otimes \xi)\] is bounded. Indeed, since \(1\otimes \xi=1\otimes \xi+0=0+1\otimes \xi\), we have \[\begin{align} \gamma(1\otimes \xi, 1\otimes \xi) &\leq s_\Phi(1\otimes \xi, 1\otimes \xi)+ s_\Psi(0, 0) \\ &\leq \|\Phi\|\|\xi\|^2. \end{align}\] Similarly, \[\gamma(1\otimes \xi, 1\otimes \xi) \leq \|\Psi\|\|\xi\|^2.\] Hence, \[\gamma(1\otimes \xi, 1\otimes \xi) \leq\min\{\|\Phi\|, \|\Psi\|\}\|\xi\|^2,\] which implies that \(V\) is bounded. Then we define the CP map \(\Theta\) on \(\mathcal{M}\) by \[\Theta(x)=V^*xV\] for \(x\in\mathcal{M}\).
We finally show that \(\Theta(x)\in\mathcal{N}\) for \(x\in\mathcal{M}\). By (2 ), for \(b\in\mathcal{N}\), we have \[\mathopen{\langle} \Theta(x)Jb^*J\xi, \eta \mathclose{\rangle} = \mathopen{\langle} Jb^*J\Theta(x)\xi, \eta \mathclose{\rangle}.\] Therefore \(\Theta(x)\in(\mathcal{N}')'=\mathcal{N}\). By (3 ), we also have \[\mathopen{\langle} \Theta(y^*x)\xi, \eta \mathclose{\rangle} = (s_\Phi : s_\Psi)(x\otimes \xi, y\otimes \eta).\]
Definition 2. For \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), the parallel sum \(\Phi : \Psi\) is defined by \[\Phi : \Psi \mathrel{\vcenter{:}}= \Theta.\]
Therefore, the norm estimate and other basic properties are summarized as follows.
Proposition 9. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then the following properties hold:
\(\Phi : \Psi = \Psi : \Phi\),
\(\Phi : \Phi = \frac{1}{2}\Phi\),
\(\Phi : \Psi\leq_{\mathrm{cp}}\Phi\) and \(\Phi : \Psi\leq_{\mathrm{cp}}\Psi\). Inparticular, \(\|\Phi : \Psi\| \leq \min\{\|\Phi\|, \|\Psi\|\}\).
Before stating the fundamental properties of the parallel sum, we provide its block matrix characterization. As with the geometric mean, the corresponding characterization for CP maps follows directly from Proposition 3 (1).
Proposition 10. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then \[\Phi : \Psi = \max \left\{ \Theta \in \mathrm{CP}(\mathcal{M}, \mathcal{N}) \;\middle|\; \begin{bmatrix} \Phi & \Phi \\ \Phi & \Phi + \Psi \\ \end{bmatrix} \geq_{\mathrm{cp}} \begin{bmatrix} \Theta & 0 \\ 0 & 0 \\ \end{bmatrix} \right\}.\]
By employing the same arguments in the case of the geometric mean, we can also establish analogous fundamental properties for the parallel sum.
Theorem 4. Let \(\Phi, \Psi, \Phi_i, \Psi_i \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) for \(i=1,2\). The following properties hold:
\(\Phi_1 \leq_{\mathrm{cp}} \Phi_2\) and \(\Psi_1 \leq_{\mathrm{cp}} \Psi_2\) imply \((\Phi_1 : \Psi_1) \leq_{\mathrm{cp}} (\Phi_2 : \Psi_2)\),
For any von Neumann algebra \(\mathcal{P}\) and \(\Xi \in \mathrm{CP}(\mathcal{N}, \mathcal{P})\), \[\Xi \circ (\Phi : \Psi) \leq_{\mathrm{cp}} (\Xi \circ \Phi) : (\Xi \circ \Psi),\]
For any von Neumann algebra \(\mathcal{L}\) and \(\Xi \in \mathrm{CP}(\mathcal{L}, \mathcal{M})\), \[(\Phi : \Psi) \circ \Xi \leq_{\mathrm{cp}} (\Phi \circ \Xi) : (\Psi \circ \Xi),\]
\(\Phi_n \downarrow \Phi\) and \(\Psi_n \downarrow \Psi\) imply \(\Phi_n : \Psi_n \downarrow \Phi : \Psi\).
Corollary 2. Let \(\Phi_1, \Phi_2 \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Suppose that \(\alpha \in \mathrm{Aut}(\mathcal{N})\) and \(\beta \in \mathrm{Aut}(\mathcal{M})\). Then \[\alpha \circ (\Phi_1 : \Phi_2) \circ \beta = (\alpha \circ \Phi_1 \circ \beta) : (\alpha \circ \Phi_2 \circ \beta).\]
Proposition 11. Let \(\Phi, \Psi, \Phi_i, \Psi_j\in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) for \(i=1,2\). Then \[(\Phi_1 : \Psi_1)+(\Phi_2 : \Psi_2)\leq(\Phi_1+\Phi_2) : (\Psi_1+\Psi_2).\]
Finally, to bridge to the next section, we introduce the harmonic mean for CP maps. In the Kubo-Ando theory, the harmonic mean is simply a scalar multiple of the parallel sum. We adopt the same definition here.
Definition 3. For \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\), the harmonic mean \(\Phi\, !\, \Psi\) is defined by \[\Phi\, !\, \Psi \mathrel{\vcenter{:}}= 2 (\Phi : \Psi).\]
The block matrix characterization of the harmonic mean also follows from Proposition 3 (2).
Proposition 12. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then \[\Phi\, !\, \Psi = \max \left\{ \Theta \in \mathrm{CP}(\mathcal{M}, \mathcal{N}) \;\middle|\; \begin{bmatrix} 2\Phi & 0 \\ 0 & 2\Psi \\ \end{bmatrix} \geq_{\mathrm{cp}} \begin{bmatrix} \Theta & \Theta \\ \Theta & \Theta \\ \end{bmatrix} \right\}.\]
In this section, we establish the Arithmetic-Geometric-Harmonic Mean (AM-GM-HM) inequality for CP maps.
Definition 4. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). The arithmetic mean \(\Phi\nabla\Psi\) is naturally defined by \[\Phi\nabla\Psi\mathrel{\vcenter{:}}=\frac{\Phi + \Psi}{2}.\]
This approach highlights the intrinsic order-theoretic nature of the AM–GM–HM inequality and provides a perspective compatible with the connection framework developed later.
Theorem 5 (AM-GM-HM inequality). Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then \[\Phi\, !\, \Psi \leq_{\mathrm{cp}} \Phi \# \Psi \leq_{\mathrm{cp}} \Phi\nabla\Psi.\]
Proof. By [4], the associated forms \(s_\Phi\) and \(s_\Psi\) can be represented by compatible representations, \[s_\Phi(\zeta_1, \zeta_2) = \mathopen{\langle}Ah(\zeta_1), h(\zeta_2)\mathclose{\rangle} \quad \text{and} \quad s_\Psi(\zeta_1, \zeta_2) = \mathopen{\langle}B h(\zeta_1), h(\zeta_2)\mathclose{\rangle}.\] By the definition of the operator means for positive sesquilinear forms, the forms \(s_{\Phi\, !\, \Psi}\), \(s_{\Phi \# \Psi}\), and \(s_{\Phi \nabla \Psi}\) correspond exactly to the operators \(A\, !\, B\), \(A \# B\), and \(A\nabla B\), respectively. The Kubo-Ando theory guarantees that \[A\, !\, B \leq A \# B \leq A\nabla B.\] Taking the inner product with \(h(\zeta)\) immediately yields that \[s_{\Phi\, !\, \Psi}\leq s_{\Phi \# \Psi} \leq s_{\Phi \nabla \Psi}.\] Since the correspondence \(\Theta \mapsto s_\Theta\) preserves the order, we finally conclude that \[\Phi\, !\, \Psi \leq_{\mathrm{cp}} \Phi \# \Psi \leq_{\mathrm{cp}} \Phi \nabla \Psi.\] ◻
Corollary 3. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then the following conditions are equivalent:
\(\Phi \nabla \Psi = \Phi \# \Psi\),
\(\Phi \# \Psi = \Phi\, !\, \Psi\),
\(\Phi = \Psi\).
Proof. It is trivial that (3) implies both (1) and (2).
To prove that (1) implies (3), suppose that \(\Phi \nabla \Psi = \Phi \# \Psi\). The corresponding bounded positive operators \(A, B\) constructed in the proof of Theorem 5 must satisfy \[A\nabla B = A \# B.\] By the Kubo-Ando theory, the equality condition for the AM-GM inequality holds if and only if \(A = B\). Thus, we have \(A = B\), which implies that \(s_\Phi= s_\Psi\). Hence, we conclude that \(\Phi = \Psi\).
The implication from (2) to (3) follows by exactly the same argument using the equality condition for the GM-HM inequality. ◻
Corollary 4. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) be unital. Then \[(\Phi\, !\, \Psi)(1) \leq (\Phi \# \Psi)(1) \leq 1.\]
Proof. Evaluating the AM-GM-HM inequality at the identity \(1 \in \mathcal{M}\), we obtain \[(\Phi\, !\, \Psi)(1) \leq (\Phi \# \Psi)(1) \leq (\Phi \nabla \Psi)(1) = \frac{\Phi(1) + \Psi(1)}{2} = \frac{1 + 1}{2} = 1.\] ◻
In this section, we consider the finite-dimensional case to illustrate our main results. Let \(\mathcal{M}= M_m(\mathbb{C})\) and \(\mathcal{N}= M_n(\mathbb{C})\). We denote by \(\mathop{\mathrm{Tr}}\) the canonical non-normalized trace. Recall that every CP map \(\Phi \colon M_m(\mathbb{C}) \to M_n(\mathbb{C})\) is uniquely represented by its Choi matrix \(C_\Phi \in M_m(\mathbb{C}) \otimes M_n(\mathbb{C})\) defined by \[C_\Phi \mathrel{\vcenter{:}}= \sum_{i,j=1}^m e_{ij} \otimes \Phi(e_{ij}),\] where \(\{e_{ij}\}_{i,j=1}^m\) is the standard matrix units for \(M_m(\mathbb{C})\). This correspondence is called the Choi–Jamiołkowski isomorphism. The map \(\Phi \mapsto C_\Phi\) is a affine isomorphism that preserves the order, i.e., \(\Phi \leq_{\mathrm{cp}} \Psi\) if and only if \(C_\Phi \leq C_\Psi\) as positive semi-definite matrices. Using this correspondence, the geometric mean of CP maps naturally reduces to the geometric mean of their Choi matrices.
Proposition 13. Let \(\Phi, \Psi \colon M_m(\mathbb{C}) \to M_n(\mathbb{C})\) be CP maps, and let \(C_\Phi, C_\Psi \in M_m(\mathbb{C}) \otimes M_n(\mathbb{C})\) be their Choi matrices. Then, the Choi matrix \(C_{\Phi \# \Psi}\) of the geometric mean \(\Phi \# \Psi\) is exactly the geometric mean of \(C_\Phi\) and \(C_\Psi\), that is, \[C_{\Phi \# \Psi} = C_\Phi \# C_\Psi.\]
Proof. Let \(\Theta \colon M_m(\mathbb{C}) \to M_n(\mathbb{C})\) be a CP map. By Proposition 7, \(\Theta \leq_{\mathrm{cp}} \Phi \# \Psi\) if and only if the block map \[\Pi = \begin{bmatrix} \Phi & \Theta \\ \Theta & \Psi \\ \end{bmatrix} \colon M_m(\mathbb{C}) \to M_2(M_n(\mathbb{C}))\] is CP. Then a direct computation of the corresponding Choi matrix shows that \[C_\Pi = \begin{bmatrix} C_\Phi & C_\Theta \\ C_\Theta & C_\Psi \\ \end{bmatrix}.\] Since \(\Pi\) is CP if and only if \(C_\Pi \geq 0\), we have that \(\Theta \leq_{\mathrm{cp}} \Phi \# \Psi\) is equivalent to \[\begin{bmatrix} C_\Phi & C_\Theta \\ C_\Theta & C_\Psi \\ \end{bmatrix} \geq 0.\] Recall that the geometric mean \(C_\Phi \# C_\Psi\) is defined as the maximum positive matrix \(X\) satisfying \[\begin{bmatrix} C_\Phi & X \\ X & C_\Psi \end{bmatrix} \geq 0.\] Because the Choi–Jamiołkowski isomorphism \(\Theta \mapsto C_\Theta\) is order-preserving, the maximal CP map \(\Phi \# \Psi\) corresponds exactly to the maximal matrix \(C_\Phi \# C_\Psi\). Therefore, we conclude that \(C_{\Phi \# \Psi} = C_\Phi \# C_\Psi\). ◻
Remark 14. By using the block matrix characterization for the harmonic mean (Proposition 12) and applying exactly the same argument as in Proposition 13, we can easily deduce that the Choi matrix of the harmonic mean is the harmonic mean of the respective Choi matrices: \[C_{\Phi\, !\, \Psi} = C_\Phi\, !\, C_\Psi.\] By the Choi–Jamiołkowski isomorphism, the AM-GM-HM inequality for CP maps (Theorem 5) in the finite-dimensional case perfectly translates to the classical AM-GM-HM inequality for positive matrices: \[C_\Phi\, !\, C_\Psi \leq C_\Phi \# C_\Psi \leq C_\Phi \nabla C_\Psi.\] This correspondence ensures that our generalizations of the operator means to CP maps are fully consistent with the classical matrix theory via the Choi–Jamiołkowski isomorphism.
Remark 15. Let \(\Phi,\Psi \colon M_m(\mathbb{C})\to M_n(\mathbb{C})\) be CP maps with \[C_\Phi = rP, \qquad C_\Psi = sQ\] for some projections \(P, Q\) and scalar \(r,s\geq 0\). Then \[C_\Phi\#C_\Psi=\sqrt{rs}(P\wedge Q) \quad\text{and}\quad C_\Phi\, !\, C_\Psi=\frac{2rs}{r+s}(P\wedge Q).\]
Indeed, for projections \(P,Q\), it is well-known that \(P\# Q=P\, !\, Q=P\wedge Q\). Therefore, we obtain \[C_\Phi\# C_\Psi=(rC_\Phi)\#(sC_\Psi)=\sqrt{rs}(P\# Q)= \sqrt{rs}(P\wedge Q),\] and \[C_\Phi\, !\, C_\Psi=(rC_\Phi)\, !\, (sC_\Psi)=\frac{2rs}{r+s}(P\, !\, Q)=\frac{2rs}{r+s}(P\wedge Q).\]
Example 1 (Non-unitality). We note that even if \(\Phi\) and \(\Psi\) are unital CP maps, their geometric mean \(\Phi \# \Psi\) need not be unital.
Let \(\mathcal{M} = \mathcal{N} = M_2(\mathbb{C})\). Consider the identity map \(\Phi(x)=x\) and the unitary conjugation \[\Psi(x)=UxU^*,\qquad U= \begin{bmatrix} 1 & 0 \\ 0 & -1 \\ \end{bmatrix}.\] Clearly, \(\Phi(1)=1\) and \(\Psi(1)=1\). Their Choi matrices \(C_\Phi\), \(C_\Psi\) are \[C_\Phi=\sum_{i,j=1}^2 e_{ij}\otimes e_{ij}, \quad C_\Psi=\sum_{i,j=1}^2 e_{ij}\otimes (Ue_{ij}U^*).\] Let \(\{e_i\}_{i=1,2}\) be the canonical orthonormal basis of \(\mathbb{C}^2\). Note that \(e_{ij}=e_ie_j^*\). Define \[v_\Phi=e_1\otimes e_1+e_2\otimes e_2, \quad v_\Psi=e_1\otimes e_1-e_2\otimes e_2 \in\mathbb{C}^2\otimes\mathbb{C}^2 .\] Then \[C_\Phi=v_\Phi v_\Phi^*, \qquad C_\Psi=v_\Psi v_\Psi^*,\] and so both Choi matrices are positive rank-one operators. Let \(P\) and \(Q\) be the projections onto \(\mathbb{C}v_\Phi\) and \(\mathbb{C}v_\Psi\), respectively. Then \[C_\Phi=\|v_\Phi\|^2 P, \qquad C_\Psi=\|v_\Psi\|^2 Q.\] Since \(\langle v_\Phi, v_\Psi \rangle = 0\), we have \(v_\Phi\perp v_\Psi\). Therefore, \(\mathrm{ran}(P)\cap\mathrm{ran}(Q)=\{0\}\). By Remark 15, we obtain \(\Phi\#\Psi=0\).
Example 2 (Density matrices). Let \(\mathcal{M}=M_n(\mathbb{C})\) and \(\mathcal{N}=\mathbb{C}\). A positive functional on \(M_n(\mathbb{C})\) is a CP map \(\varphi \colon M_n(\mathbb{C})\to\mathbb{C}\). Every positive functional \(\varphi\) is represented by a density matrix \(\rho\in M_n(\mathbb{C})^+\) such that for \(x\in M_n(\mathbb{C})\), \[\varphi(x)=\mathrm{Tr}(\rho x).\] Similarly, let \(\psi\) be another positive functional represented by a density matrix \(\sigma\in M_n(\mathbb{C})\).
We compute the Choi matrix of \(\varphi\). Since \(M_n(\mathbb{C})\otimes\mathbb{C}\) is naturally identified with \(M_n(\mathbb{C})\), we obtain \[C_\varphi = \sum_{i,j=1}^n e_{ij}\otimes\varphi(e_{ij}) = \sum_{i,j=1}^n \mathrm{Tr}(\rho e_{ij}) e_{ij} = \sum_{i,j=1}^n \rho_{ji} e_{ij} = \rho^T .\] Thus, the Choi matrix of a positive functional is the transpose of its density matrix. Similarly, we have \(C_\psi=\sigma^T\).
Now we consider the geometric mean \(\varphi \#\psi\). Since the transpose map \(A\mapsto A^T\) is an order-preserving isomorphism on positive semidefinite matrices, it commutes with operator means. Hence, \[C_{\varphi \#\psi} = C_\varphi \# C_\psi = \rho^T \#\sigma^T = (\rho \#\sigma)^T .\] By the Choi–Jamiołkowski isomorphism, we conclude that \[(\varphi \#\psi)(x) =\mathrm{Tr}((\rho \#\sigma)x).\] Therefore, the geometric mean of positive functionals corresponds exactly to the geometric mean of their density matrices.
Example 3 (Identity channel and completely depolarizing channel).
In quantum information theory, two fundamental extremal channels are the identity channel, representing perfect transmission of quantum information, and the completely depolarizing channel, representing maximal noise. We compute the geometric and harmonic means of these two channels.
Let \(\mathcal{M}=\mathcal{N}=M_d(\mathbb{C})\) with \(d\ge2\). Consider the identity channel \[\mathrm{id}(x)=x\] and the completely depolarizing channel \[\Delta(x)=\frac{\mathrm{Tr}(x)}{d}\, 1 .\] Both maps are unital CP.
Let \(C_{\mathrm{id}},C_{\Delta}\in M_d(\mathbb{C})\otimes M_d(\mathbb{C})\) denote their Choi matrices. They are given by \[C_{\mathrm{id}}=\sum_{i,j=1}^d e_{ij}\otimes e_{ij}, \qquad C_{\Delta} =\sum_{i,j=1}^d e_{ij}\otimes\Delta(e_{ij}) =\sum_{i=1}^d e_{ii}\otimes\frac{1}{d}1 =\frac{1}{d}(1\otimes1).\] Let \[v=\sum_{i=1}^d e_i\otimes e_i\in\mathbb{C}^d\otimes\mathbb{C}^d .\] Then \[C_{\mathrm{id}}=vv^*\] is a rank-one positive operator. In contrast, \[C_\Delta=\frac{1}{d} I\] is a scalar multiple of the identity on \(\mathbb{C}^d\otimes\mathbb{C}^d\). Let \(P\) be the projection onto \(\mathbb{C}v\), and \(Q=I\). Then \[C_{\mathrm{id}}= dP,\qquad C_\Delta=\frac{1}{d}Q.\] Since \(\mathrm{ran}(P)\cap\mathrm{ran}(Q)=\mathrm{ran}(P)\), by Remark 15, we have \[C_{\mathrm{id}}\# C_\Delta =\sqrt{d\cdot\frac{1}{d}}(P\wedge Q)=P=\frac{1}{d}C_{\mathrm{id}},\] and \[C_{\mathrm{id}}\, !\, C_\Delta =\frac{2dd^{-1}}{d+d^{-1}}(P\wedge Q)=\frac{2d}{d^2+1}P=\frac{2}{d^2+1}C_{\mathrm{id}}.\] Thus the geometric mean corresponds to the projection onto the maximally entangled state.
By the Choi-Jamiołkowski isomorphism, the resulting quantum channels satisfy \[\mathrm{id} \# \Delta=\frac{1}{d}\, \mathrm{id}, \qquad \mathrm{id}\, !\, \Delta=\frac{2}{d^2+1}\, \mathrm{id}.\]
Example 4 (Schur multipliers). Let \(\mathcal{M}=\mathcal{N}=M_n(\mathbb{C})\). For \(A\in M_n(\mathbb{C})\), the Schur multiplier \(S_A\colon M_n(\mathbb{C})\to M_n(\mathbb{C})\) is defined by \[S_A(x)=A\circ x,\] where \(\circ\) denotes the entrywise (Hadamard) product. It is well-known that \(S_A\) is CP if and only if \(A\) is positive semidefinite.
We compute the Choi matrix of \(S_A\). Using the canonical matrix units \(\{e_{ij}\}_{i,j=1}^n\), we obtain \[\begin{align} C_{S_A} &= \sum_{i,j=1}^n e_{ij}\otimes S_A(e_{ij}) \\ &= \sum_{i,j=1}^n e_{ij}\otimes (A_{ij}e_{ij}) \\ &=\sum_{i,j=1}^n A_{ij}(e_i e_j^*\otimes e_i e_j^*). \end{align}\] Observe that \[e_i e_j^*\otimes e_i e_j^* = (e_i\otimes e_i)(e_j\otimes e_j)^* .\] Let \(V \colon \mathbb{C}^n\to\mathbb{C}^n\otimes\mathbb{C}^n\) be the isometry defined by \[V e_i = e_i\otimes e_i\] for \(i=1,\dots,n\). Then the above expression can be written as \[C_{S_A}=VAV^* .\]
Now let \(A, B\ge 0\). Since \(V\) is an isometry, the transformer inequality for operator means yields \[(VAV^*) \# (VBV^*) = V(A \# B)V^* .\] Therefore, \[C_{S_A}\# C_{S_B} = V(A\#B)V^* .\] Recognizing that \(V(A\#B)V^*\) is precisely the Choi matrix of the Schur multiplier \(S_{A\#B}\), we conclude that \[S_A\# S_B = S_{A\#B}.\]
Example 5 (Adjoint maps). Let \(\mathcal{M}= \mathcal{N}= M_2(\mathbb{C})\). For \(A \in M_2(\mathbb{C})\), we define a CP map \[\Psi_A(x) = A x A^*\] for \(x \in M_2(\mathbb{C})\). Let \[A = \begin{bmatrix} 2 & 0 \\ 0 & 1 \\ \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 0 \\ 0 & 2 \\ \end{bmatrix}.\] Since \(A\) and \(B\) commute, their geometric mean is \[C = A \# B = A^{1/2} B^{1/2} = \begin{bmatrix} \sqrt{2} & 0 \\ 0 & \sqrt{2} \\ \end{bmatrix} = \sqrt{2}\,I.\] Hence \[\Psi_C(x) = (\sqrt{2}I)x(\sqrt{2}I)^* = 2x.\]
We now compute the geometric mean \(\Psi_A\# \Psi_B\). Their Choi matrices are rank-one operators \[C_{\Psi_A}=v_Av_A^*, \qquad C_{\Psi_B}=v_Bv_B^*,\] where \[v_A = \begin{bmatrix} 2 \\ 0 \\ 0 \\ 1 \\ \end{bmatrix}, \qquad v_B = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 2 \\ \end{bmatrix}.\] Let \(P\) and \(Q\) be the projections onto \(\mathbb{C}v_A\) and \(\mathbb{C}v_B\), respectively. Then we have \[C_{\Psi_A}=\|v_A\|^2 P, \qquad C_{\Psi_B}=\|v_B\|^2 Q.\] Since \(v_A, v_B\) are linearly independent, \(\mathrm{ran}(P)\cap\mathrm{ran}(Q)=\{0\}\). By Remark 15, we obtain \[C_{\Psi_A} \# C_{\Psi_B}=(\|v_A\|^2 P)\#(\|v_B\|^2 Q) =\|v_A\|\|v_B\|(P\wedge Q)=0.\] Consequently, \[\Psi_A \# \Psi_B = 0.\] Therefore, in general, \[\Psi_A \# \Psi_B \neq \Psi_{A\# B}.\] Thus, for adjoint CP maps, the geometric mean does not behave functorially with respect to the operator geometric mean of the implementing operators.
Remark 16. For general CP maps \(\Phi\) and \(\Psi\) with Kraus representations \[\Phi(x) = \sum_{i} A_i x A_i^* \quad\text{and}\quad \Psi(x) = \sum_{j} B_j x B_j^*,\] it is non-trivial to compute an explicit Kraus representation for their geometric mean \(\Phi \# \Psi\). Their Choi matrices are given by \[C_\Phi = \sum_{i} v_{A_i} v_{A_i}^* \quad\text{and}\quad C_\Psi = \sum_{j} v_{B_j} v_{B_j}^*.\] Since \[C_\Phi \# C_\Psi = C_\Phi^{1/2}(C_\Phi^{-1/2}C_\Psi C_\Phi^{-1/2})^{1/2}C_\Phi^{1/2},\] if \(C_\Phi\) and \(C_\Psi\) are invertible, there is no simple formula to express the Kraus operators of \(\Phi \# \Psi\) in terms of \(A_i\) and \(B_j\).
However, the spatial structure of the geometric mean is described by operator mean theory. As shown by [25], if \(\mathrm{ran}(A)\) and \(\mathrm{ran}(B)\) are closed, then \[\mathrm{ran}(A \# B) = \mathrm{ran}(A) \cap \mathrm{ran}(B).\] Hence we obtain \[\mathrm{ran}(C_{\Phi \# \Psi}) = \mathrm{span}\{v_{A_i}\}_i \cap \mathrm{span}\{v_{B_j}\}_j.\]
Let \((\mathcal{M}, \mathcal{H}, J, \mathcal{P})\) be a standard form of a von Neumann algebra \(\mathcal{M}\). Let \(\varphi\) and \(\psi\) be normal positive functionals on \(\mathcal{M}\). To define their geometric mean, we consider the dominating functional \[\omega \mathrel{\vcenter{:}}= \varphi + \psi .\] We denote its a unique implementing vector \(\Omega \in \mathcal{P}\). Since \(\varphi \leq \omega\) and \(\psi \leq \omega\), there exist unique positive operators \(h', k' \in \mathcal{M}'\) such that \[\varphi(x) = \langle x h' \Omega, \Omega \rangle, \qquad \psi(x) = \langle x k' \Omega, \Omega \rangle\] for \(x \in \mathcal{M}\). Note that \(h'+k'=1_{\mathcal{H}}\), in paticular, \(h'\) and \(k'\) commute.
Thus the Pusz–Woronowicz geometric mean \(\varphi \# \psi\) is given by \[(\varphi \# \psi)(x) = \langle x (h' \# k') \Omega, \Omega \rangle = \langle x \sqrt{h'k'}\, \Omega, \Omega \rangle\] for \(x \in \mathcal{M}\).
Remark 17. It is instructive to compare the Pusz–Woronowicz mean with another interpolation of states introduced in [8], [26], which is based on Connes’ relative modular operator \(\Delta_{\varphi,\psi}\).
We show that these two constructions produce different states in the noncommutative setting by computing them explicitly in the matrix case \(\mathcal{M}=M_n(\mathbb{C})\).
Let \(\varphi(x)=\mathrm{Tr}(\rho x)\) and \(\psi(x)=\mathrm{Tr}(\sigma x)\) for invertible positive definite matrices \(\rho\) and \(\sigma\), respectively We use the standard Hilbert space \(\mathcal{H}=M_n(\mathbb{C})\) equipped with the Hilbert–Schmidt inner product \[\langle x,y\rangle=\mathrm{Tr}(y^*x).\] The representative vectors in \(\mathcal{P}\) are \[\Omega_\varphi=\rho^{1/2}, \qquad \Omega_\psi=\sigma^{1/2}.\] The relative modular operator is given by \[\Delta_{\varphi,\psi}(x)=\rho x\sigma^{-1},\] and \[\Delta_{\varphi,\psi}^{1/4}(x) = \rho^{1/4}x\sigma^{-1/4} \quad\text{for}\;x\in\mathcal{H}.\] In particular, \[\Delta_{\varphi,\psi}^{1/4}\Omega_\psi = \rho^{1/4}\sigma^{1/4}.\]
The resulting state is \[\omega_{1/2}(x) = \mathrm{Tr}\!\left( (\rho^{1/4}\sigma^{1/4})^* x (\rho^{1/4}\sigma^{1/4}) \right) = \mathrm{Tr}(\rho^{1/4}\sigma^{1/2}\rho^{1/4}x).\] In particular, \[\omega_{1/2}(1) = \mathrm{Tr}(\rho^{1/2}\sigma^{1/2}).\] On the other hand, \[\mathrm{Tr}(\rho\#\sigma) = \mathrm{Tr}\!\left( \rho^{1/2} (\rho^{-1/2}\sigma\rho^{-1/2})^{1/2} \rho^{1/2} \right).\]
These quantities are naturally related to the Uhlmann fidelity in quantum information theory, defined by \[F(\rho,\sigma) = \mathrm{Tr}\!\sqrt{\rho^{1/2}\sigma\rho^{1/2}}.\]
It is known that \[\mathrm{Tr}(\rho\#\sigma) \le \mathrm{Tr}(\rho^{1/2}\sigma^{1/2}) \le \mathrm{Tr}\sqrt{\rho^{1/2}\sigma\rho^{1/2}},\] and equality holds if and only if \(\rho\) and \(\sigma\) commute (see [27], [28]).
Conditional expectations are fundamental objects in the theory of von Neumann algebras, particularly in subfactor theory and quantum probability. It is therefore natural to investigate how operator means interact with conditional expectations.
Let \(\mathcal{N}\subseteq \mathcal{M}\) be an inclusion of von Neumann algebras. Recall that a projection from \(\mathcal{M}\) onto \(\mathcal{N}\) is a linear map \(E \colon \mathcal{M}\to \mathcal{N}\) such that \(E(a)=a\) for every \(a\in\mathcal{N}\). A conditional expectation from \(\mathcal{M}\) onto \(\mathcal{N}\) is a contractive CP projection \(E \colon \mathcal{M}\to \mathcal{N}\) such that \(E(a x b)=aE(x)b\) for \(a\), \(b\in \mathcal{N}\), \(x\in \mathcal{M}\). A normal conditional expectation preserving a fixed faithful normal state is unique by Takesaki’s theorem [29].
Let \[E_i \colon \mathcal{M}\to \mathcal{N}_i\] be conditional expectations for \(i=1,2\). Composing with the natural inclusions \(j_i \colon \mathcal{N}_i \hookrightarrow \mathcal{M}\), we regard \[\tilde{E}_i \mathrel{\vcenter{:}}= j_i \circ E_i\] as CP maps on \(\mathcal{M}\).
Proposition 18. Let \(E_i \colon \mathcal{M}\to\mathcal{N}_i\) be conditional expectations for \(i=1,2\) and set \[\Theta \mathrel{\vcenter{:}}= \tilde{E}_1 \# \tilde{E}_2 \in \mathrm{CP}(\mathcal{M},\mathcal{M}).\] Then the following hold:
\(\Theta\) is an \((\mathcal{N}_1\cap \mathcal{N}_2)\)-bimodule map, i.e., \[\Theta(a x b)=a\Theta(x)b\] for \(a,b\in \mathcal{N}_1\cap\mathcal{N}_2,\;x\in \mathcal{M}\),
\(\Theta\) is sub-unital and \[\Theta(1)\in (\mathcal{N}_1\cap\mathcal{N}_2)' \cap \mathcal{M}.\]
Proof. (1) We first show that \(\Theta\) intertwines inner automorphisms implemented by elements in \(\mathcal{N}_1 \cap \mathcal{N}_2\), and then extend the result by linearity. Let \(a\in \mathcal{N}_1\cap\mathcal{N}_2\) be invertible. We define \(\Psi_a(x)=axa^*\).
Since \[\begin{bmatrix} \tilde{E}_1 & \Theta \\ \Theta & \tilde{E}_2 \end{bmatrix} \ge_{\mathrm{cp}}0,\] we have \[\begin{bmatrix} \Psi_a\circ \tilde{E}_1 & \Psi_a\circ\Theta\\ \Psi_a\circ\Theta & \Psi_a\circ \tilde{E}_2 \end{bmatrix} \ge_{\mathrm{cp}}0.\] Since \(a \in \mathcal{N}_i\) and \(E_i\) is a conditional expectation onto \(\mathcal{N}_i\), we have \(E_i(axa^*) = a E_i(x) a^*\), i.e., \[\tilde{E}_i \circ \Psi_a = \Psi_a \circ \tilde{E}_i.\] Then we obtain \[\begin{bmatrix} \tilde{E}_1\circ\Psi_a & \Psi_a\circ\Theta\\ \Psi_a\circ\Theta & \tilde{E}_2\circ\Psi_a \end{bmatrix} \ge_{\mathrm{cp}}0.\] Therefore, \[\begin{bmatrix} \tilde{E}_1 & \Psi_a\circ\Theta\circ\Psi_{a^{-1}}\\ \Psi_a\circ\Theta\circ\Psi_{a^{-1}} & \tilde{E}_2 \end{bmatrix} \ge_{\mathrm{cp}}0.\] By the maximality property of the geometric mean with respect to the CP order, we obtain \[\Psi_a\circ\Theta\circ\Psi_{a^{-1}} \le_{\mathrm{cp}} \Theta,\] and thus, \[\Psi_a\circ\Theta \le_{\mathrm{cp}} \Theta\circ\Psi_a .\] Exchanging \(a\) and \(a^{-1}\) yields the reverse inequality, so \[\Theta\circ\Psi_a=\Psi_a\circ\Theta.\] Thus \[\Theta(axa^*)=a\Theta(x)a^* .\]
For unitaries \(u, v\in \mathcal{N}_1\cap\mathcal{N}_2\) and \(z\in\mathbb{C}\), we put \(a=u+zv=u(1+zu^*v)\). If \(|z|<1\), then \(\|zu^*v\|<1\), and so \(1+zu^*v\) is invertible. Thanks to \(\Theta(axa^*)=a\Theta(x)a^*\), we can obtain \[\Theta(vxu^*)=v\Theta(x)u^*.\] For any \(a, b\in \mathcal{N}_1\cap\mathcal{N}_2\), using the standard fact that every element in a C\(^*\)-algebra is a linear combination of unitaries, we conclude \[\Theta(axb)=a\Theta(x)b.\]
(2) By Corollary 4, the geometric mean of two unital CP maps is sub-unital, hence \(\Theta(1)\le1\).
Applying the bimodule property to \(x=1\) gives \[a\Theta(1)=\Theta(a)=\Theta(1)a \quad\text{for}\;a\in \mathcal{N}_1\cap\mathcal{N}_2,\] which shows \[\Theta(1)\in (\mathcal{N}_1\cap\mathcal{N}_2)' \cap \mathcal{M}.\] ◻
We can similarly prove the following:
Corollary 5. Let \(E_1, E_2 \colon \mathcal{M}\to \mathcal{N}\) be conditional expectations. Then
\(E_1\# E_2\) is an \(\mathcal{N}\)-bimodule map,
\(E_1\# E_2\) is sub-unital such that \[(E_1\# E_2)(1) \in \mathcal{N}' \cap \mathcal{M}.\]
The notion of index for CP maps was introduced by Pimsner and Popa [16], and further studied in [15]. The Pomsner-Popa index \(\mathrm{Ind}_{\mathrm{cp}}(\Phi)\) of \(\Phi\in\mathrm{CP}(\mathcal{M}, \mathcal{M})\) is defined by \[\mathrm{Ind}_{\mathrm{cp}}(\Phi)\mathrel{\vcenter{:}}= \inf\{\lambda>0\mid \lambda\Phi-\mathop{\mathrm{id}}_{\mathcal{M}}\geq_{\mathrm{cp}} 0\}.\]
Proposition 19. Let \(\Phi_i\in\mathrm{CP}(\mathcal{M}, \mathcal{M})\) for \(i=1,2\). If there exist \(\lambda_i>0\), then \[\Phi_i \geq_{\mathrm{cp}} \lambda_i\;\mathrm{id}_{\mathcal{M}} \quad\text{for}\;i=1,2,\] then \[\Phi_1 \# \Phi_2 \geq_{\mathrm{cp}} \sqrt{\lambda_1 \lambda_2}\;\mathrm{id}_\mathcal{M}.\] In particular, if \(\mathrm{Ind}_{\mathrm{cp}}(\Phi_i)<\infty\) for \(i=1,2\), then \[\mathrm{Ind}_{\mathrm{cp}}(\Phi_1\#\Phi_2)\leq\sqrt{\mathrm{Ind}_{\mathrm{cp}}(\Phi_1)\,\mathrm{Ind}_{\mathrm{cp}}(\Phi_2)}\]
Proof. Since \(\Phi_i \geq_{\mathrm{cp}} \lambda_i \mathrm{id}_{\mathcal{M}}\) for \(i=1,2\), by Theorem 3, we have \[\Phi_1 \# \Phi_2 \geq_{\mathrm{cp}} (\lambda_1 \mathrm{id}_{\mathcal{M}}) \# (\lambda_2 \mathrm{id}_{\mathcal{M}}) = \sqrt{\lambda_1 \lambda_2} \, \mathrm{id}_{\mathcal{M}}.\] Hence, the statement holds. ◻
Example 6. Let \(\mathcal{N}_1 = \mathcal{N}_2 = M_n(\mathbb{C})\), and \(\mathcal{M}=\mathcal{N}_1\otimes\mathcal{N}_2\). Let \(\psi_1\), \(\psi_2\) be states on \(\mathcal{N}_1\), \(\mathcal{N}_2\) given by \[\psi_1(x) = \mathrm{Tr}(\rho x), \quad\psi_2(y) = \mathrm{Tr}(\sigma y)\] with diagonal invertible density matrices \(\rho=\mathrm{diag}(\rho_1,\dots,\rho_n)\), \(\sigma=\mathrm{diag}(\rho_1,\dots,\rho_n)\), respectively. Then we define conditional expectations \(E_1\), \(E_2\) from \(\mathcal{M}\) onto \(\mathcal{N}_1\), \(\mathcal{N}_2\) by \[E_1(x_1\otimes x_2)=x_1\otimes \psi_2(x_2)1, \quad E_2(x_1\otimes x_2)=\psi_1(x_1)1\otimes x_2,\] respectively.
We consider the geometric mean \(\tilde{E}_1\# \tilde{E}_2\). Now, we see the corresponding Choi matrices. \[\begin{align} C_{\tilde{E}_1} &= \sum_{i,j,k,l}(e_{ij}\otimes e_{kl})\otimes \tilde{E}_1(e_{ij}\otimes e_{kl}) \\ &= \sum_{i,j,k,l}(e_{ij}\otimes e_{kl})\otimes (e_{ij}\otimes \psi_2(e_{kl})1) \\ &= (\sum_{i,j}e_{ij}\otimes e_{ij})\otimes (\sum_ke_{kk}\otimes \sigma_k1) \\ &= C_{\mathrm{id}_{\mathcal{N}_1}}\otimes\boldsymbol{\sigma}, \end{align}\] where \(\boldsymbol{\sigma}\) be the \(n^2 \times n^2\) diagonal matrix defined by \[\boldsymbol{\sigma} = \bigoplus_{k=1}^n \sigma_k I_n\] so that its diagonal entries are \((\underbrace{\sigma_1, \dots, \sigma_1}_{n}, \dots, \underbrace{\sigma_n, \dots, \sigma_n}_{n})\). Similarly, we have \[C_{\tilde{E}_2}=\boldsymbol{\rho}\otimes C_{\mathrm{id}_{\mathcal{N}_2}}.\] Notice that \[\mathopen{\langle}v, A^{-1}v\mathclose{\rangle}^{-1}=\max\{\lambda>0 \mid A-\lambda vv^*\geq 0 \}\] for an invertible positive definite matrix \(A\) and a non-zero vector \(v\). Note that \[\begin{align} 0&\leq C_{\tilde{E}_1}-C_{\lambda\mathop{\mathrm{id}}_{\mathcal{M}}} \\ &= C_{\mathrm{id}_{\mathcal{N}_1}}\otimes\boldsymbol{\sigma}-\lambda C_{\mathrm{id}_{\mathcal{N}_1}}\otimes C_{\mathrm{id}_{\mathcal{N}_2}} \\ &= C_{\mathrm{id}_{\mathcal{N}_1}}\otimes(\boldsymbol{\sigma}-\lambda C_{\mathrm{id}_{\mathcal{N}_2}}). \end{align}\] Recall that \(C_{\mathrm{id}}=vv^*\), where \[v=\sum_i e_i\otimes e_i.\] Hence \[\lambda_\sigma\mathrel{\vcenter{:}}=\mathrm{Ind}_{\mathrm{cp}}(\tilde{E}_1)^{-1}=(\sum_i\sigma_i^{-1})^{-1} = \mathopen{\langle} v, \boldsymbol{\sigma}^{-1}v \mathclose{\rangle}^{-1}.\] Therefore \[\tilde{E}_1\geq_{\mathrm{cp}}\lambda_\sigma\mathop{\mathrm{id}}_{\mathcal{M}}.\] Similarly, \[\tilde{E}_2\geq_{\mathrm{cp}}\lambda_\rho\mathop{\mathrm{id}}_{\mathcal{M}}.\] By Proposition 6, we have \[\tilde{E}_1\# \tilde{E}_2\geq_{\mathrm{cp}}\sqrt{\lambda_\rho\lambda_\sigma}\;\mathrm{id}_{\mathcal{M}}.\] Moreover, by using Proposition 13 and [30], we have \[\begin{align} C_{\tilde{E}_1\#\tilde{E}_2} &= C_{\tilde{E}_1}\#C_{\tilde{E}_2} \\ &= (C_{\mathrm{id}_{\mathcal{N}_1}}\otimes\boldsymbol{\sigma})\#(\boldsymbol{\rho}\otimes C_{\mathrm{id}_{\mathcal{N}_2}}) \\ &= (C_{\mathrm{id}_{\mathcal{N}_1}}\#\boldsymbol{\rho})\otimes(\boldsymbol{\sigma}\# C_{\mathrm{id}_{\mathcal{N}_2}}). \end{align}\] Since \[\boldsymbol{\rho}^{-1/2}C_{\mathrm{id}}\boldsymbol{\rho}^{-1/2} = (\boldsymbol{\rho}^{-1/2}v)(\boldsymbol{\rho}^{-1/2}v)^*,\] we have \[(\boldsymbol{\rho}^{-1/2}C_{\mathrm{id}_{\mathcal{N}_1}}\boldsymbol{\rho}^{-1/2})^{1/2} =\frac{(\boldsymbol{\rho}^{-1/2}v)(\boldsymbol{\rho}^{-1/2}v)^*}{\sqrt{\sum_{i=1}^n\rho_i^{-1}}}.\] Therefore \[C_{\mathrm{id}_{\mathcal{N}_1}}\#\boldsymbol{\rho} = \boldsymbol{\rho}^{1/2}(\boldsymbol{\rho}^{-1/2}C_{\mathrm{id}_{\mathcal{N}_1}}\boldsymbol{\rho}^{-1/2})^{1/2}\boldsymbol{\rho}^{1/2} = \frac{1}{\sqrt{\sum_{i=1}^n\rho_i^{-1}}}C_{\mathrm{id}_{\mathcal{N}_1}}.\] Similarly, \[\boldsymbol{\sigma}\# C_{\mathrm{id}_{\mathcal{N}_2}} = \frac{1}{\sqrt{\sum_{i=1}^n\sigma_i^{-1}}}C_{\mathrm{id}_{\mathcal{N}_2}}.\] Then we obtain \[C_{\tilde{E}_1\#\tilde{E}_2} = \sqrt{\lambda_\rho\lambda_\sigma}\;(C_{\mathrm{id}_{\mathcal{N}_1}}\otimes C_{\mathrm{id}_{\mathcal{N}_2}}) = \sqrt{\lambda_\rho\lambda_\sigma}\;C_{\mathrm{id}_{\mathcal{M}}}.\] Therefore \[\tilde{E}_1\#\tilde{E}_2=\sqrt{\lambda_\rho\lambda_\sigma}\;\mathrm{id}_{\mathcal{M}},\] which implies \[\mathrm{Ind}_{\mathrm{cp}}(\tilde{E}_1\#\tilde{E}_2) = (\sqrt{\lambda_\rho\lambda_\sigma})^{-1} = \sqrt{(\sum_{i=1}^n\rho_i^{-1})(\sum_{i=1}^n\sigma_i^{-1})}.\]
Example 7. Let \(\mathcal{M}= M_2(\mathbb{C})\supset \mathcal{N}_1=\mathbb{C}^2\) (diagonal), and \(E_1\colon\mathcal{M}\to\mathcal{N}_1\) be the canonical conditional expectation. Set \[u= \begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \\ \end{bmatrix},\] and \(\mathcal{N}_2=u\mathcal{N}_1 u^*\). Then we have the conditional expectation \(E_2\colon\mathcal{M}\to\mathcal{N}_2\) by \[E_2(x)=u E_1(u^*xu)u^* \quad\text{for}\;x\in\mathcal{M}.\] To compute the geometric mean \(\tilde{E}_1\#\tilde{E}_2\), we consider the corresponding Choi matrices. One can check that \[C_{\tilde{E}_1} = \sum_{i}e_{ii}\otimes e_{ii} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}\] is the projection onto \(\mathcal{V}_1=\mathrm{span}\{e_1\otimes e_1, e_2\otimes e_2\}\), and \(C_{\tilde{E}_2}=(u\otimes u)C_{\tilde{E}_1}(u^*\otimes u^*)\) is the projection onto \(\mathcal{V}_2=\mathrm{span}\{ue_1\otimes ue_1, ue_2\otimes ue_2\}\). Note that \[\mathcal{V}_1\cap\mathcal{V}_2= \begin{cases} \mathrm{span}\{e_1\otimes e_1+e_2\otimes e_2\} & (\sin(2\theta)\ne 0), \\ \mathcal{V}_1 & (\sin(2\theta)= 0). \end{cases}\] Let \(\Theta=\tilde{E}_1\#\tilde{E}_2\). Since \(C_{\tilde{E}_1}\) and \(C_{\tilde{E}_2}\) are projections, we obtain \[C_{\Theta} = C_{\tilde{E}_1} \# C_{\tilde{E}_2}= C_{\tilde{E}_1} \wedge C_{\tilde{E}_2}.\] If \(\sin(2\theta)\ne 0\), then \[C_\Theta=\frac{1}{\|v\|^2}vv^*=\frac{1}{2}\sum_{i,j}e_{ij}\otimes e_{ij}=\frac{1}{2}C_{\mathop{\mathrm{id}}_\mathcal{M}},\] where \(v=e_1\otimes e_1+e_2\otimes e_2\). Therefore, by the Choi–Jamiołkowski isomorphism, we have \[\tilde{E}_1\# \tilde{E}_2 = \begin{cases} \frac{1}{2}\mathrm{id}_\mathcal{M}& (\sin(2\theta)\ne 0),\\ \tilde{E}_1 & (\sin(2\theta)= 0). \\ \end{cases}\]
In this section, we introduce a notion of connections for CP maps. Our goal is to extend the Kubo–Ando theory of operator connections to CP maps in a way that is compatible with their natural order structure, composition, and tensor product operations.
We begin by recalling the notion of connections for bounded positive operators on an infinite-dimensional complex Hilbert space \(\mathcal{H}\) in the sense of Kubo–Ando [1].
Definition 5. A binary operation \(\sigma : \mathbb{B}(\mathcal{H})^+\times \mathcal{B}(\mathcal{H})^+\to \mathbb{B}(\mathcal{H})^+\) is called an operator connection if it satisfies
\(A\leq C\) and \(B\leq D\) imply \(A\sigma B\leq C \sigma D\),
\(C(A\sigma B)C\leq (CAC)\sigma (CBC)\),
\(A_n\downarrow A\) and \(B_n\downarrow B\) imply \(A_n\sigma B_n\downarrow A\sigma B\) in the strong operator topology.
An operator connection \(\sigma\) is called an operator mean if
Definition 6. We denote by \(OM^+([0,\infty))\) the set of all operator monotone functions \(f\colon[0,\infty)\to[0,\infty)\).
Theorem 6 (Kubo–Ando). For each operator connection \(\sigma\), there exists a unique \(f\in OM^+([0,\infty))\) such that \[f(t)I=I\sigma (tI)\] for \(t\geq 0\). Moreover, the map \(\sigma\mapsto f\) is an affine order-isomorphism such that \(\sigma\) is an operator mean if and only if \(f(1)=1\). In this case, \(A\sigma A=A\) for all \(A\in \mathbb{B}(\mathcal{H})^+\).
By Löwner theory, for each \(f\in OM^+([0,\infty))\), there exist unique \(a, b\geq 0\) and unique finite positive measure \(\mu\) on \((0,\infty)\) such that \[\label{eq:monotone} f(t)=a+bt+\int_{(0,\infty)} \frac{t(1+\lambda)}{t+\lambda}\,d\mu(\lambda).\tag{4}\] Then the corresponding operator connection \(\sigma_f\) is given by \[A\sigma_f B=aA+bB+\int_{(0,\infty)} \frac{1+\lambda}{\lambda}\Big[(\lambda A) : B\Big]\,d\mu(\lambda).\] Moreover, for \(\alpha, \beta\in F_+(\mathcal{V})\), we define the form connection by \[\alpha\sigma_f \beta(x,y) \mathrel{\vcenter{:}}= a\alpha(x,y)+b\beta(x,y)+\int_{(0,\infty)} \frac{1+\lambda}{\lambda}\Big[(\lambda \alpha) : \beta\Big](x,y)\,d\mu(\lambda).\] It is easy to see that \(\alpha\sigma_f \beta\in F_+(\mathcal{V})\).
We refer to [11] for the notion of connections for positive sesquilinear forms, formulated in terms of (possibly unbounded) quadratic forms. While their setting differs from ours, it provides a useful conceptual background for the development of connections for CP maps.
Let \(f\in OM^+([0,\infty))\). For \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), we consider the corresponding positive sesquilinear forms \(s_\Phi, s_\Psi\) on \(\mathcal{M}\odot L^2(\mathcal{N})\). Then we set \[\gamma\mathrel{\vcenter{:}}= s_\Phi\sigma _fs_\Psi.\] Via separation and completion of \(\mathcal{M}\odot L^2(\mathcal{N})\) with respect to \(\gamma\), we obtain the Hilbert space \(\mathcal{H}_\gamma\). Then \(\mathcal{H}_\gamma\) is equipped with commuting the left action of \(\mathcal{M}\) and the right action of \(\mathcal{N}\). For \(\xi\in L^2(\mathcal{N})\), we have \[\begin{align} & \int_{(0,\infty)} \frac{1+\lambda}{\lambda}\Big[(\lambda s_\Phi) : s_\Psi\Big](1\otimes \xi, 1\otimes \xi) \,d\mu(\lambda) \\ &\quad\leq \int_{(0,1]} \frac{1+\lambda}{\lambda}\lambda s_\Phi(1\otimes \xi, 1\otimes \xi) \,d\mu(\lambda) + \int_{[1,\infty)} \frac{1+\lambda}{\lambda}s_\Psi(1\otimes \xi, 1\otimes \xi) \,d\mu(\lambda) \\ &\quad\leq 2(\|\Phi\|+\|\Psi\|) \|\mu\| \|\xi\| \end{align}\] Therefore, \[\begin{align} \gamma(1\otimes \xi, 1\otimes \xi) &= as_\Phi(1\otimes \xi, 1\otimes \xi)+bs_\Psi(1\otimes \xi, 1\otimes \xi)\\ &\quad+\int_{(0,\infty)} \frac{1+\lambda}{\lambda}[(\lambda s_\Phi) : s_\Psi](1\otimes \xi, 1\otimes \xi)\,d\mu(\lambda) \\ &\leq \Big\{a\|\Phi\|+b\|\Psi\|+2(\|\Phi\|+\|\Psi\|) \|\mu\|\Big\} \|\xi\|. \end{align}\] Hence, the map \[V : L^2(\mathcal{N})\to H_\gamma, \quad V(\xi)=h(1\otimes\xi)\] extends to a bounded operator. We define the cp map \[\Theta(x)=V^*xV\] for \(x\in M\). Moreover, one can check that \(\gamma\) satisfies (2 ). Hence, for \(b\in\mathcal{N}\), \[\mathopen{\langle} \Theta(x)Jb^*J\xi, \eta \mathclose{\rangle} = \mathopen{\langle} Jb^*J\Theta(x)\xi, \eta \mathclose{\rangle}.\] Therefore, \[\Theta(x)\in (N')'=N,\] which implies \(\Theta\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\). By (3 ), we also have \[\mathopen{\langle} \Theta(y^*x)\xi, \eta \mathclose{\rangle} = (s_\Phi \sigma_f s_\Psi)(x\otimes \xi, y\otimes \eta).\]
Definition 7. For \(f\in OM^+([0,\infty))\) given in (4 ), the cp map connection \(\sigma=\sigma_f\) of \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\) is defined by \[\Phi\sigma_f\Psi\mathrel{\vcenter{:}}=\Theta.\]
Since \(\gamma\) also satisfies (3 ), we obtain \[s_{\Phi\sigma_f\Psi}=\gamma=s_\Phi\sigma_f s_\Psi.\] We remark that if \(\Phi\), \(\Psi\) are normal, then so is \(\Phi\sigma_f\Psi\).
Proposition 20. Let \(f_1, f_2\in OM^+([0,\infty))\). Then \(\Phi\sigma_{f_1}\Psi\leq_{\mathrm{cp}} \Phi\sigma_{f_2}\Psi\) for all \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\) if and only if \(f_1(t)\leq f_2(t)\) for all \(t\geq 0\).
Proof. We realize \(s_\Phi, s_\Psi\) on the Hilbert space \(\mathcal{H}_\gamma\) with respect to \(\gamma=s_\Phi+s_\Psi\) as \[s_\Phi(x\otimes \xi, y\otimes \eta)=\langle Ah(x\otimes \xi), h(y\otimes \eta)\rangle,\] and \[s_\Phi(x\otimes \xi, y\otimes \eta)=\langle Bh(x\otimes \xi), h(y\otimes \eta)\rangle.\] Then one checks that \[s_\Phi\sigma s_\Psi(\zeta, \zeta) = \langle (A\sigma B)h(\zeta),h(\zeta)\rangle.\] Hence the assertion follows from the operator case. ◻
Since the parallel sum satisfies the following properties, one can check that the general cp map connection also holds.
Theorem 7. Let \(\sigma\) be a cp map connection and \(f\) be the corresponding function.
\(\Phi_1\leq_{\mathrm{cp}}\Phi_2\) and \(\Psi_1\leq_{\mathrm{cp}}\Psi_2\) imply \(\Phi_1\sigma\Psi_1\leq_{\mathrm{cp}}\Phi_2\sigma\Psi_2\),
For \(\Xi\in\mathrm{CP}(N, P)\), \[\Xi\circ(\Phi\sigma_f\Psi) \leq_{\mathrm{cp}} (\Xi\circ\Phi)\sigma(\Xi\circ\Psi),\]
For \(\Xi\in\mathrm{CP}(L, M)\), \[(\Phi\sigma\Psi)\circ\Xi \leq_{\mathrm{cp}} (\Phi\circ\Xi)\sigma(\Psi\circ\Xi),\]
\(\Phi_n\downarrow\Phi\) and \(\Psi_n\downarrow\Psi\) imply \(\Phi_n\sigma\Psi_n\downarrow\Phi\sigma\Psi\), where where \(\Phi_n \downarrow \Phi\) means that \(\Phi_1 \geq_{\mathrm{cp}} \Phi_2 \geq_{\mathrm{cp}} \cdots\) and \(\Phi_n(x) \to \Phi(x)\) in the ultraweak operator topology for each \(x \in \mathcal{M}^+\).
If \(f(1)=1\), then \(\Phi\sigma\Phi=\Phi\).
Corollary 6. Let \(\sigma\) be a cp map connection and \(f\) be the corresponding function.
For \(\alpha\in\mathrm{Aut}(\mathcal{N})\) and \(\beta\in\mathrm{Aut}(\mathcal{M})\), \[\alpha\circ(\Phi\sigma\Psi)\circ\beta = (\alpha\circ\Phi\circ\beta)\sigma(\alpha\circ\Psi\circ\beta),\]
\[(\Phi_1\sigma\Psi_1)+(\Phi_2\sigma\Psi_2) \leq_{\mathrm{cp}} (\Phi_1+\Phi_2)\sigma(\Psi_1+\Psi_2).\]
Example 8.
\(0\leq\alpha\leq 1\), let \(\#_\alpha\) denote the \(\alpha\)-power mean, which is the cp map mean corresponding to the operator monotone function \(t^\alpha\). Note that \(\Phi\#_0\Psi=\Phi\), \(\Phi\#_1\Psi=\Psi\) and \(\Phi\#_{1/2} \Psi=\Phi\#\Psi\).
The cp map mean corresponding to the operator monotone function \((t-1)/\log t\) is called the logarithmic mean and denoted by \(\Phi\lambda\Psi\).
We give another proof of Theorem 5.
Corollary 7. For \(\Phi, \Psi\in\mathrm{CP}(M, N)\), \[\Phi\, !\, \Psi \leq_{\mathrm{cp}} \Phi\, \#\, \Psi \leq_{\mathrm{cp}} \Phi\, \lambda\, \Psi \leq_{\mathrm{cp}} \Phi\, \nabla\, \Psi.\]
Proof. It immediately follows from the inequality \[\frac{2t}{1+t}\leq t^{1/2}\leq\frac{t-1}{\log t}\leq\frac{1+t}{2}\] for \(t>0\). ◻
Theorem 8. Let \(\Phi_k, \Psi_k\in\mathrm{CP}(\mathcal{M}_k,\mathcal{N}_k)\) for \(k=1,2\). Then \[(\Phi_1\otimes \Phi_2)\#_\alpha(\Psi_1\otimes \Psi_2) = (\Phi_1\#_\alpha\Psi_1)\otimes(\Phi_2\#_\alpha\Psi_2)\]
Proof. Set \[\gamma_k=s_{\Phi_k}+s_{\Psi_k}\] for \(k=1,2\). By the standard construction via separation and completion, we have Hilbert spaces \(\mathcal{H}_{\gamma_k}\) and the canonical maps \(h_k\colon \mathcal{M}_k\odot L^2(\mathcal{N}_k)\to \mathcal{H}_{\gamma_k}\). Then we have bounded operators \(A_k, B_k\) on \(\mathcal{H}_{\gamma_k}\) such that \[s_{\Phi_k}(x_k\otimes\xi_k, y_k\otimes \eta_k) = \mathopen{\langle} A_kh_k(x_k\otimes\xi_k), h_k(y_k\otimes \eta_k) \mathclose{\rangle}_{\gamma_k},\] and \[s_{\Psi_k}(x_k\otimes\xi_k, y_k\otimes \eta_k) = \mathopen{\langle} B_kh_k(x_k\otimes\xi_k), h_k(y_k\otimes \eta_k) \mathclose{\rangle}_{\gamma_k}.\]
Let \(\Phi=\Phi_1\otimes \Phi_2\) and \(\Psi=\Psi_1\otimes \Psi_2\). We define a sesquilinear form \(\gamma\) on \((\mathcal{M}_1 \odot L^2(\mathcal{N}_1)) \odot (\mathcal{M}_2 \odot L^2(\mathcal{N}_2))\) by \[\gamma(\zeta_1 \otimes \zeta_2, \zeta'_1 \otimes \zeta'_2) = \gamma_1(\zeta_1, \zeta'_1)\gamma_2(\zeta_2, \zeta'_2)\] for \(\zeta_1, \zeta_1'\in\mathcal{M}_1\odot L^2(\mathcal{N}_1)\) and \(\zeta_2, \zeta_2'\in\mathcal{M}_2\odot L^2(\mathcal{N}_2)\). Via separation and completion, we have a Hilbert space \(\mathcal{H}_{\gamma}\). Here we implicitly use the canonical identification \[(\mathcal{M}_1\odot L^2(N_1))\odot (\mathcal{M}_2\odot L^2(\mathcal{N}_2))=(\mathcal{M}_1\odot \mathcal{M}_2)\odot (L^2(\mathcal{N}_1)\odot L^2(\mathcal{N}_2))\] and then \(\mathcal{H}_\gamma\) can be identified with \(\mathcal{H}_{\gamma_1}\otimes \mathcal{H}_{\gamma_2}\).
We set \(h=h_1\otimes h_2\) and \(A=A_1\otimes A_2\), \(B=B_1\otimes B_2\). Then it is straightforward to verify that \(s_\Phi \sim (h,A)\) and \(s_\Psi \sim (h,B)\). Moreover, since \(A_k\) and \(B_k\) commute for each \(k\), \(A\) and \(B\) also commute. Let \[x=x_1\otimes x_2,\quad y=y_1\otimes y_2\in\mathcal{M}_1\otimes\mathcal{M}_2\] and \[\xi=\xi_1\otimes\xi_2,\quad \eta=\eta_1\otimes\eta_2\in L^2(\mathcal{N}_1)\otimes L^2(\mathcal{N}_2).\] By Pusz–Woronowicz theory, \[\begin{align} & (s_\Phi\#_\alpha s_\Psi)(x\otimes \xi, y\otimes \eta) \\ &= \mathopen{\langle} A^{1-\alpha} B^\alpha h(x\otimes \xi), h(y\otimes \eta) \mathclose{\rangle}_\gamma \\ &= \mathopen{\langle} A_1^{1-\alpha} B_1^\alpha h_1(x_1\otimes \xi_1), h_1(y_1\otimes \eta_1) \mathclose{\rangle}_{\gamma_1} \mathopen{\langle} A_2^{1-\alpha} B_2^\alpha h_2(x_2\otimes \xi_2), h_2(y_2\otimes \eta_2) \mathclose{\rangle}_{\gamma_2} \\ &= (s_{\Phi_1}\#_\alpha s_{\Psi_1})(x_1\otimes \xi_1, y_1\otimes \eta_1) (s_{\Phi_2}\#_\alpha s_{\Psi_2})(x_2\otimes \xi_2, y_2\otimes \eta_2) \\ &= (s_{\Phi_1\#_\alpha \Psi_1}\otimes s_{\Phi_2\#_\alpha\Psi_2})(x\otimes \xi, y\otimes \eta). \end{align}\] Therefore, \[s_{\Phi\#_\alpha \Psi}=s_{\Phi_1\#_\alpha \Psi_1}\otimes s_{\Phi_2\#_\alpha\Psi_2}.\] Since the associated sesquilinear forms coincide, we conclude that \[\Phi\#_\alpha \Psi = (\Phi_1\#_\alpha \Psi_1)\otimes(\Phi_2\#_\alpha \Psi_2).\] ◻
Definition 8. Let \(\sigma\) be a cp map mean and \(f\) be the corresponding function.
The cp map mean with the representing function \(tf(t^{-1})\) is called the transpose of \(\sigma\) and denoted by \(\sigma'\). If \(\sigma=\sigma'\), then \(\sigma\) is said to be symmetric.
The cp map mean with the representing function \(f(t^{-1})^{-1}\) is called the adjoint of \(\sigma\) and denoted by \(\sigma^*\).
The cp map mean with the representing function \(t/f(t)\) is called the dual of \(\sigma\) and denoted by \(\sigma^\perp\).
The following propositions are verified from the definitions and the proof of operator case. These correspond exactly to the classical notions in Kubo–Ando theory.
Proposition 21. Let \(\sigma\) be a cp map mean and \(f\) be the corresponding function.
\(\Phi\sigma\Psi=\Psi\sigma'\Phi\)
\((\sigma')'=\sigma\), \((\sigma^*)^*=\sigma\), and \((\sigma^\perp)^\perp=\sigma\).
\(\sigma^\perp=(\sigma')^*=(\sigma^*)'\), \(\sigma'=(\sigma^*)^\perp=(\sigma^\perp)^*\), and \(\sigma^*=(\sigma')^\perp=(\sigma^\perp)'\).
Proposition 22. If \(\sigma\) is a symmetric mean, then \(!\leq_{\mathrm{cp}}\sigma\leq_{\mathrm{cp}}\nabla\).
Proposition 23. For every cp map mean \(\sigma\), the following hold:
\((\Phi\sigma\Psi)+(\Psi\sigma\Phi)\leq_{\mathrm{cp}}\Phi+\Psi\),
\((\Phi\sigma\Psi)\, :\, (\Psi\sigma\Phi)\geq_{\mathrm{cp}}\Phi\, :\, \Psi\),
\((\Phi\sigma\Psi)\#(\Phi\sigma^\perp\Psi)=(\Phi\#\Psi)\),
\((\Phi\sigma\Psi)+(\Phi\sigma^\perp\Psi)\leq_{\mathrm{cp}}\Phi+\Psi\),
\((\Phi\sigma\Psi)\, :\, (\Phi\sigma^\perp\Psi)\geq_{\mathrm{cp}} (\Phi\, :\, \Psi)\).
Remark 24. In the present paper, we pay particular attention to the geometric mean \(\sharp\) corresponding to \(f(t) = \sqrt{t}\). The primary reason for this focus is its unparalleled symmetry. While the arithmetic mean \(\nabla\) and the harmonic mean \(!\) are symmetric (i.e., \(\nabla = \nabla'\) and \(! = !'\)), they are mutually adjoint (\(\nabla^* = !\)) and dual (\(\nabla^\perp = !\)). Conversely, the geometric mean is identically self-adjoint (\(\sharp = \sharp^*\)) and self-dual (\(\sharp = \sharp^\perp\)). Thus, \(\sharp\) serves as the unique invariant fixed point under all the fundamental transformations established in the propositions above.
In this section, we establish a Lebesgue-type decomposition for CP maps. Our approach is inspired by the classical theory for positive operators developed by Ando [2] and the form-theoretic framework of Simon [31], and is further informed by subsequent refinements due to Kosaki [3], [32], [33] and, more recently, Aibara and Ueda [34].
Related Radon–Nikodym type results for CP maps originate from the work of Arveson [17] and have been further studied in various contexts, including quantum operations and covariant CP maps (see, e.g., [35], [36]).
The aim of this section is to establish such a decomposition within our framework, based on the parallel sum operation introduced in the previous sections.
More precisely, we introduce notions of absolute continuity and singularity for CP maps, and show that every CP map admits a decomposition into an absolutely continuous part and a singular part with respect to a fixed reference map.
First, we introduce the notions of absolute continuity and singularity for CP maps based on the parallel sum. In what follows, the limits of CP maps are taken in the point-ultraweak operator topology.
Definition 9. Let \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\).
\(\Psi\) is said to be \(\Phi\)-absolutely continuous, denoted by \(\Psi \ll \Phi\), if there exists a sequence of CP maps \(\{\Psi_n\}\) and positive real numbers \(\{\alpha_n\}\) such that \[\Psi_n \uparrow \Psi \quad \text{and} \quad \Psi_n \leq_{\mathrm{cp}} \alpha_n \Phi \quad\text{for all}\;n.\]
\(\Psi\) is said to be \(\Phi\)-singular, denoted by \(\Phi \perp \Psi\), if for any \(\Theta\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\), \[\Theta \leq_{\mathrm{cp}} \Phi \quad \text{and} \quad \Theta \leq_{\mathrm{cp}} \Psi \implies \Theta = 0.\]
To constructively prove the Lebesgue decomposition, the parallel sum operation provides a powerful algebraic characterization of these order-theoretic definitions.
Let \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\). Set \(\Psi_n \mathrel{\vcenter{:}}= (n\Phi \, : \, \Psi)\). By Proposition 9 and Theorem 4, we have \(\Psi_n \leq \Psi\) and the sequence \(\{\Psi_n\}\) is monotonically increasing. Hence there exists the point-ultraweak limit \[[\Phi]\Psi \mathrel{\vcenter{:}}= \lim_{n \to \infty} (n\Phi : \Psi)\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\] such that \([\Phi]\Psi\leq_\mathrm{cp}\Psi\). Since \(\Psi_n \leq n\Phi\) by Proposition 9, setting \(\alpha_n = n\), the sequence \(\{\Psi_n\}\) satisfies the condition for \([\Phi]\Psi \ll \Phi\).
This provides a canonical choice of the absolutely continuous component. For convenience, we also write \[\Psi_{\mathrm{ac}} := [\Phi]\Psi.\]
Lemma 25. Let \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\).
\(\Phi \perp \Psi\) if and only if \(\Phi \, : \, \Psi = 0\).
\(\Psi \ll \Phi\) if and only if \(\Psi =[\Phi]\Psi\).
Proof. (1) Suppose \(\Theta \leq_{\mathrm{cp}} \Phi\) and \(\Theta \leq_{\mathrm{cp}} \Psi\). By the monotonicity of the parallel sum (Theorem 4), \[\frac{1}{2}\Theta = \Theta \, : \, \Theta \leq_{\mathrm{cp}} \Phi \, : \, \Psi.\] Hence, if \(\Phi \, : \, \Psi = 0\), then \(\Theta = 0\), which means \(\Phi \perp \Psi\). Conversely, by Proposition 9, we have \[\Phi \, : \, \Psi \leq_{\mathrm{cp}} \Phi \quad\text{and} \quad \Phi \, : \, \Psi \leq_{\mathrm{cp}} \Psi.\] If \(\Phi \perp \Psi\), then it immediately follows that \(\Phi \, : \, \Psi = 0\).
(2) If \[\Psi = \lim_{n \to \infty} (n\Phi : \Psi),\] then \(\Psi \ll \Phi\).
Conversely, assume that \(\Psi \ll \Phi\). Then there exists a sequence of CP maps \(\{\Psi_m\}\) and positive real numbers \(\{\alpha_m\}\) such that \[\Psi_m \uparrow \Psi \quad \text{and} \quad \Psi_m \leq_{\mathrm{cp}} \alpha_m \Phi \quad\text{for all}\;m.\]
Now we claim that \[\Psi_m-(n\Phi : \Psi_m) \leq_{\mathrm{cp}} \frac{\alpha_m}{n+\alpha_m}\Psi.\] Consider a sesquilinear form \(\gamma=s_\Phi+s_\Psi\). Then there exist compatible representations \(s_\Phi\sim(h, A)\) and \(s_\Psi\sim(h,B)\) on the Hilbert space \(\mathcal{H}_\gamma\). Then, since \(\Psi_m\leq_{\mathrm{cp}}\Psi\), there exist bounded operators \(B_m\) such that \[s_{\Psi_m}(x\otimes\xi, y\otimes\eta) = \mathopen{\langle} B_mh(x\otimes\xi), h(y\otimes\eta) \mathclose{\rangle}.\] Hence we obtain \(B_m\uparrow B\) and \(B_m\leq\alpha_m A\), i.e., \(B\) is \(A\)-absolutely continuous. By the proof of [2], we have \[B_m-(nA : B_m)\leq\frac{\alpha_m}{n+\alpha_m}B.\] Therefore, we conclude our claim.
Taking the limit as \(n \to \infty\), we obtain \[\Psi_m=\lim_{n\to\infty}(n\Phi : \Psi_m),\] which means \(\Psi_m\ll\Phi\) by the above. Since \[\Psi_m= \lim_{n\to\infty}(n\Phi : \Psi_m) \leq_{\mathrm{cp}} \lim_{n\to\infty}(n\Phi : \Psi) \leq_{\mathrm{cp}}\Psi,\] taking the limit as \(m \to \infty\), we have \[\Psi=\lim_{n\to\infty}(n\Phi : \Psi)=[\Phi]\Psi.\] ◻
Theorem 9. Let \(\Phi, \Psi\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\). Then there exists a decomposition \[\Psi = \Psi_{\mathrm{ac}} + \Psi_{\mathrm{s}}\] such that \(\Psi_{\mathrm{ac}}, \Psi_{\mathrm{s}}\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\) satisfying \(\Psi_{\mathrm{ac}}\ll \Phi\) and \(\Psi_{\mathrm{s}} \perp \Phi\). This decomposition is not unique in general. However, there exists a canonical maximum choice for the absolutely continuous part. Specifically, the maximum among all \(\Theta\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\) such that \(\Theta \leq_{\mathrm{cp}} \Psi\) and \(\Theta \ll \Phi\) is given by \[\Psi_{\mathrm{ac}} = \lim_{n \to \infty} (n\Phi \, : \,\Psi).\] Furthermore, this Lebesgue decomposition is unique if and only if there exists a constant \(\alpha > 0\) such that \(\Psi_{\mathrm{ac}} \leq_{\mathrm{cp}} \alpha \Phi\).
Proof. Recall that \(\Psi_{\mathrm{ac}} \ll \Phi\). We then define the remaining part as \[\Psi_{\mathrm{s}} \mathrel{\vcenter{:}}= \Psi - \Psi_{\mathrm{ac}}\in \mathrm{CP}(\mathcal{M}, \mathcal{N}).\] Consider a sesquilinear form \(\gamma=s_\Phi+s_\Psi\). Then there exist compatible representations \(s_\Phi\sim(h, A)\) and \(s_\Psi\sim(h,B)\) on the Hilbert space \(\mathcal{H}_\gamma\). Note that \[\mathopen{\langle} (n\Phi\, :\, \Psi)(y^*x)\xi, \eta \mathclose{\rangle} = \gamma((nA : B)h(x\otimes \xi), h(y\otimes \eta)).\] Hence \[\mathopen{\langle} \Psi_{\mathrm{ac}}(y^*x)\xi, \eta \mathclose{\rangle} = \gamma([A]Bh(x\otimes \xi), h(y\otimes \eta)),\] where \[[A]B=\lim_{n\to\infty}(nA):B.\] Therefore \[\mathopen{\langle} \Psi_{\mathrm{s}}(y^*x)\xi, \eta \mathclose{\rangle} = \gamma((B-[A]B)h(x\otimes \xi), h(y\otimes \eta)),\] By [2], the corresponding operator \(B-[A]B\) is \(A\)-singular. If \(\Theta\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\) such that \(\Theta \leq_{\mathrm{cp}}\Phi\) and \(\Theta \leq_{\mathrm{cp}} \Psi_{\mathrm{s}}\), then there exists a positive operator \(D\) such that \[\mathopen{\langle} \Theta(y^*x)\xi, \eta \mathclose{\rangle} = \gamma(Dh(x\otimes \xi), h(y\otimes \eta)).\] Since \(D\leq A\), \(D\leq B-[A]B\), we have \(D=0\), which means \(\Theta=0\). Therefore \(\Psi_{\mathrm{s}}\) is \(\Phi\)-singular.
Let \(\Theta\in\mathrm{CP}(\mathcal{M}, \mathcal{N})\) such that \(\Theta \leq_{\mathrm{cp}} \Psi\) and \(\Theta \ll \Phi\). By Lemma 25, we have \[\Theta=\lim_{n\to\infty}(n\Phi : \Theta) \leq_{\mathrm{cp}}\lim_{n\to\infty}(n\Phi : \Psi) =\Psi_{\mathrm{ac}}.\]
Finally, we consider the condition for uniqueness. Suppose that there is another Lebesgue decomposition \(\Psi = \Psi_1 + \Psi_2\) where \(\Psi_1 \ll \Phi\) and \(\Psi_2 \perp \Phi\). Passing to the Hilbert space \(\mathcal{H}_\gamma\), this corresponds to a decomposition of positive operators \(B = B_1 + B_2\) such that \(B_1 \ll A\) and \(B_2 \perp A\). By [2], this decomposition is unique (meaning \(B_1 = [A]B\)) if and only if \([A]B \leq \alpha A\) for some \(\alpha\geq 0\) Translating this back to CP maps via the affine order-isomorphism, \([A]B \leq \alpha A\) holds if and only if \(\Psi_{\mathrm{ac}} \leq_{\mathrm{cp}} \alpha \Phi\). Thus, the Lebesgue decomposition for CP maps is unique if and only if \(\Psi_{\mathrm{ac}} \leq_{\mathrm{cp}} \alpha \Phi\) for some \(\alpha > 0\). ◻
Example 9 (Lebesgue decomposition of normal positive functionals). Let \((\mathcal{M}, \mathcal{H}, J, \mathcal{P})\) be a standard form of a von Neumann algebra \(\mathcal{M}\). We assume that \(\varphi\in \mathcal{M}_*^+\) is a faithful normal state. For a normal positive functional \(\psi\in\mathcal{M}_*^+\), we set \(\omega= \varphi + \psi\). We denote by \(\xi_\omega\in\mathcal{P}\) the unique implementing vector, which is cyclic and separating for \(\mathcal{M}\). Since \(\varphi \leq \omega\) and \(\psi \leq \omega\), there exist unique positive operators \(h'_\varphi, h'_\psi \in \mathcal{M}'\) such that \(0 \leq h'_\varphi, h'_\psi \leq 1\), \(h'_\varphi + h'_\psi = 1\), and \[\varphi(x) = \mathopen{\langle} h'_\varphi x\xi_\omega, \xi_\omega \mathclose{\rangle}, \quad \psi(x) = \mathopen{\langle} h'_\psi x\xi_\omega, \xi_\omega \mathclose{\rangle}\] for all \(x \in \mathcal{M}\). These operators \(h'_\varphi\) and \(h'_\psi\) correspond to our general operators \(A\) and \(B\).
Let \(e'_\varphi \mathrel{\vcenter{:}}= \mathrm{supp}(h'_\varphi)\) and \(e'_\psi \mathrel{\vcenter{:}}= \mathrm{supp}(h'_\psi)\) be the support projections of \(h'_\varphi\) and \(h'_\psi\) in \(\mathcal{M}'\), which correspond to the orthogonal projections onto \(\overline{\mathrm{ran}}(h'_\varphi)= (\ker h'_\varphi)^\perp\) and \(\overline{\mathrm{ran}}(h'_\psi)= (\ker h'_\psi)^\perp\), respectively.
Now we consider the Lebesgue decomposition of \(\psi\) with respect to \(\varphi\) via the parallel sum limit in Theorem 9. By the functional calculus in \(\mathcal{M}'\), the sequence \((n h'_\varphi : h'_\psi)\) converges strongly to \(h'_\psi e'_\varphi\).
Indeed, \(h'_\varphi\) and \(h'_\psi\) are positive operators in \(\mathcal{M}'\) satisfying \(h'_\varphi + h'_\psi = 1\). This implies that \(h'_\varphi\) and \(h'_\psi\) commute.
For each \(n \geq 1\), we define the real-valued continuous function \(f_n\) on \([0,1]\) by \[f_n(t) \mathrel{\vcenter{:}}= \frac{n t (1 - t)}{n t + (1 - t)}.\] (Here, we continuously extend \(f_n(0) = 0\)). Note that \(0 \leq f_n(t) \leq 1\) for all \(t \in [0,1]\) and all \(n\). Then we have \((n h'_\varphi : h'_\psi) = f_n(h'_\varphi)\).
Notice that the sequence \(\{f_n(t)\}\) converges pointwise to the Borel function \[f(t) \mathrel{\vcenter{:}}= \chi_{(0,1]}(t)(1 - t),\] where \(\chi_{(0,1]}\) is the characteristic function of the interval \((0,1]\).
Since \(0 \le f_n \leq 1\) and \(f_n \to f\) pointwise, the bounded convergence theorem for functional calculus implies that the sequence of operators \(f_n(h'_\varphi)\) converges strongly to the operator \(f(h'_\varphi)\). Applying the functional calculus, we get \[f(h'_\varphi) =\chi_{(0,1]}(h'_\varphi) (1 - h'_\varphi) .\] Note that the spectral projection \(\chi_{(0,1]}(h'_\varphi)\) is exactly the support projection \(e'_\varphi\) of \(h'_\varphi\). Since \(h'_\psi = 1- h'_\varphi\), we conclude that \[\mathrm{s}\text{-}\!\!\lim_{n \to \infty} (n h'_\varphi : h'_\psi) = e'_\varphi h'_\psi .\] Here, \(h'_\psi e'_\varphi=e'_\varphi h'_\psi\).
Thus, the CP map limit naturally splits the state \(\psi\) into its absolutely continuous and singular parts: \[\begin{align} \psi_{\mathrm{ac}}(x) &= \lim_{n \to \infty} (n\varphi : \psi)(x) = \langle e'_\varphi h'_\psi x \xi_\omega, \xi_\omega \rangle, \\ \psi_{\mathrm{s}}(x) &= \psi(x) - \psi_{\mathrm{ac}}(x) = \langle (1-e'_\varphi) h'_\psi x \xi_\omega, \xi_\omega \rangle, \end{align}\] which shows that the absolutely continuous part is supported on \(e'_\varphi\), while the singular part is supported on \(1-e'_\varphi\).
In [3], Kosaki explicitly constructs the maximal \(\varphi\)-absolutely continuous part \(\tilde{\psi}(\leq\psi)\) of \(\psi\), and shows \(\psi=\tilde{\psi}+(\psi-\tilde{\psi})\), where the rest \(\psi-\tilde{\psi}\) is \(\varphi\)-singular. We briefly recall his construction. Let \(a=[D\varphi : D\omega]_{-i/2}\in\mathcal{M}\). By [3], \(\ker(a)=0\) if and only if \(\psi\) is \(\varphi\)-absolutely continuous. Let \(p'\) be the projection onto \(\ker(JaJ)\). Note that \(p'=JpJ\in\mathcal{M}'\), where \(p\) is the projection onto \(\ker(a)\). Then \(\tilde{\psi}\in\mathcal{M}_*^+\) is given by \[\begin{align} \tilde{\psi}(x) &=\mathopen{\langle} (1-p')x\xi_\omega, \xi_\omega \mathclose{\rangle}-\varphi(x) \\ &= \mathopen{\langle} J\{(1-p)-a^*a\}Jx\xi_\omega, \xi_\omega \mathclose{\rangle} \end{align}\] for \(x\in\mathcal{M}\). We remark that \(\tilde{\psi}\) is indeed positive, because \(0\leq a^*a\leq 1\) and \(1-p=\mathrm{supp}(a^*a)\). Note that \[\varphi(x) = \mathopen{\langle} Ja^*aJx\xi_\omega,\xi_\omega \mathclose{\rangle}, \quad\text{and}\quad (\psi-\tilde{\psi})(x) = \mathopen{\langle} p'x\xi_\omega, \xi_\omega \mathclose{\rangle}.\] Therefore, \(Ja^*aJ=h'_\varphi\). Since \((\overline{\mathrm{ran}}(h'_\varphi))^\perp=\ker(h'_\varphi)=\ker(JaJ)\), we have \(1-p'=\mathrm{supp}(h'_\varphi)=e'_\varphi\). Moreover, \(\ker(a)=0\) if and only if \(p'=0\), i.e., \(e'_\psi\leq e'_\varphi=1\).
Moreover, since \(h'_\varphi+h'_\psi=1\), we have \[\mathopen{\langle}(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle} = \mathopen{\langle}h'_\varphi(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle} + \mathopen{\langle}h'_\psi(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle}.\] On the first term, since \(h'_\varphi(1-p')=h'_\varphi\), we obtain \[\mathopen{\langle}h'_\varphi(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle} = \mathopen{\langle}h'_\varphi x\xi_\omega, \xi_\omega\mathclose{\rangle} = \varphi(x).\] On the second term, we have \[\mathopen{\langle}h'_\psi(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle} = \mathopen{\langle}h'_\psi e'_\varphi x\xi_\omega, \xi_\omega\mathclose{\rangle} = \psi_{\mathrm{ac}}(x).\] Hence, \[\mathopen{\langle}(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle}=\varphi(x)+\psi_{\mathrm{ac}}(x).\] Therefore, we have \[\psi_{\mathrm{ac}}(x)=\mathopen{\langle}(1-p')x\xi_\omega, \xi_\omega\mathclose{\rangle}-\varphi(x)=\tilde{\psi}(x),\] and so \(\psi_{\mathrm{s}}=\psi-\tilde{\psi}\).
This shows that our Lebesgue decomposition of normal positive functionals coincides with Kosaki’s one and is recovered as a special case of our general decomposition for CP maps.
When \(\varphi\) is not faithful, let \(e \mathrel{\vcenter{:}}= s(\varphi) \in \mathcal{M}\) be the support projection of \(\varphi\). We can easily extend the above result to this general case by reducing the corner \(e\mathcal{M}e\). Let \(\varphi_e\) and \(\psi_e\) be the restrictions of \(\varphi\) and \(\psi\) to \(e\mathcal{M}e\), respectively. Since \(\varphi_e\) is faithful on \(e\mathcal{M}e\), the previous argument applies, yielding the Lebesgue decomposition \(\psi_e = (\psi_e)_{\mathrm{ac}} + (\psi_e)_{\mathrm{s}}\) on \(e\mathcal{M}e\). To recover the decomposition on the entire algebra \(\mathcal{M}\), we note that the parallel sum \((n\varphi : \psi)\) is supported on \(e\), because \(\mathrm{supp}(n\varphi : \psi) \leq \mathrm{supp}(\varphi) = e\). Thus, for any \(x \in \mathcal{M}\), since \((n\varphi : \psi)\) is supported on \(e\), we have \[(n\varphi : \psi)(x) = (n\varphi : \psi)(exe) = (n\varphi_e : \psi_e)(exe).\] Taking the limit \(n \to \infty\), we obtain \[\psi_{\mathrm{ac}}(x) =\lim_{n \to \infty}(n\varphi : \psi)(x) = (\psi_e)_{\mathrm{ac}}(exe).\] The singular part on \(\mathcal{M}\) is given by \[\psi_{\mathrm{s}}(x) = \psi(x) - \psi_{\mathrm{ac}}(x).\]
The preceding discussion reveals that the Lebesgue decomposition is naturally encoded by support projections associated with Radon–Nikodym derivatives. The following theorem shows that this geometric picture extends to arbitrary CP maps via the Stinespring representation.
Theorem 10. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\) and set \(\Gamma=\Phi+\Psi\). Let \((\pi, V, \mathcal{H}_\gamma)\) be the minimal Stinespring representation of \(\Gamma\), and let \(A', B' \in \pi(\mathcal{M})'\) be the Arveson Radon–Nikodym derivatives corresponding to \(\Phi\) and \(\Psi\), respectively.
Set \(E_\Phi = \overline{\mathrm{ran}}(A')\) and \(E_\Psi = \overline{\mathrm{ran}}(B')\), and let \(P'_\Phi, P'_\Psi \in \pi(\mathcal{M})'\) be the corresponding support projections.
Then the Lebesgue decomposition of \(\Psi\) with respect to \(\Phi\) is given by \[\Psi_{\mathrm{ac}}(x) = V^*(P'_\Phi B'P'_\Phi )\pi(x)V, \quad \Psi_{\mathrm{s}}(x) = V^*((I-P'_\Phi) B' (I-P'_\Phi) )\pi(x)V.\]
Moreover, the following equivalences hold:
\(\Psi \ll \Phi \iff E_\Psi \subseteq E_\Phi \iff P'_\Psi \le P'_\Phi\),
\(\Psi \perp \Phi \iff E_\Psi \perp E_\Phi \iff P'_\Psi P'_\Phi = 0\).
Proof. Let \(\Phi, \Psi \in \mathrm{CP}(\mathcal{M}, \mathcal{N})\). Set \(\Gamma = \Phi + \Psi\) and \(\gamma=s_\Phi+s_\Psi\). We can define a bounded linear map \(V \colon L^2(\mathcal{N}) \to \mathcal{H}_\gamma\) by \[V\xi \mathrel{\vcenter{:}}= h(1_\mathcal{M}\otimes \xi),\] and a \(*\)-representation \(\pi \colon \mathcal{M}\to \mathcal{B}(\mathcal{H}_\gamma)\) by \[\pi(a)h(x \otimes \xi) \mathrel{\vcenter{:}}= h((ax) \otimes \xi),\] where \(h\colon \mathcal{M}\odot L^2(\mathcal{N}) \to \mathcal{H}_\gamma\) is the canonical map. Then it is easily verified that the minimal Stinespring dilation of \(\Gamma\) is given by \((\pi, V, \mathcal{H}_\gamma)\), i.e., \[\Gamma(x) = V^* \pi(x) V \quad \text{for all } x \in \mathcal{M},\] and \(\mathcal{H}_\gamma = \overline{\mathrm{span}}\{ \pi(x)V\xi \mid x \in \mathcal{M}, \xi \in L^2(\mathcal{N}) \}\).
Since \(\Phi \leq_{\mathrm{cp}} \Gamma\) and \(\Psi \leq_{\mathrm{cp}} \Gamma\), Arveson’s Radon-Nikodym theorem for CP maps guarantees the existence of unique positive operators in the commutant \(\pi(\mathcal{M})'\) that implement \(\Phi\) and \(\Psi\), respectively. These operators coincide with the previously introduced positive operators. We have \(A', B' \in \pi(\mathcal{M})'\), \(A'+B'=I\), and \[\Phi(x) = V^* A' \pi(x) V, \quad \Psi(x) = V^* B' \pi(x) V.\]
Let \(E_\Phi \mathrel{\vcenter{:}}= \overline{\mathrm{ran}}(A')\) and \(E_\Psi \mathrel{\vcenter{:}}= \overline{\mathrm{ran}}(B')\). Since \(A'\) and \(B'\) belong to \(\pi(\mathcal{M})'\), the orthogonal projections \(P'_\Phi\) and \(P'_\Psi\) onto \(E_\Phi\) and \(E_\Psi\), respectively, also belong to \(\pi(\mathcal{M})'\). Because \(A'+B'=I\), the operators \(A'\) and \(B'\) commute, which implies that their support projections \(P'_\Phi\) and \(P'_\Psi\) also commute.
By similar functional calculus arguments as in Example 9, the CP map limit naturally splits \(\Psi\) into its absolutely continuous and singular parts: \[\begin{align} \Psi_{\mathrm{ac}}(x) &= V^* (P'_\Phi B'P'_\Phi)\pi(x)V, \\ \Psi_{\mathrm{s}}(x) &= V^* ((I-P_\Phi)B'(I-P'_\Phi))\pi(x)V. \end{align}\]
Recall that because the Stinespring dilation \((\pi, V, \mathcal{H}_\gamma)\) is minimal, the map \(T \mapsto V^* T \pi(\cdot) V\) is an affine and order-preserving isomorphism from the positive cone of \(\pi(\mathcal{M})'\) onto the set of CP maps dominated by a multiple of \(\Gamma\). In particular, for any positive operators \(S, T \in \pi(\mathcal{M})'\), we have \(V^* S \pi(\cdot) V = V^* T \pi(\cdot) V\) if and only if \(S = T\).
(1) By Lemma 25, \(\Psi \ll \Phi\) if and only if \(\Psi = \Psi_{\mathrm{ac}}\). From the above, this holds if and only if \[V^* B' \pi(x) V = V^* (P'_\Phi B'P'_\Phi) \pi(x) V\] for all \(x \in \mathcal{M}\). Due to the minimality of the Stinespring dilation, this is equivalent to the operator identity \(B' = P'_\Phi B'P'_\Phi\), which means \(\overline{\mathrm{ran}}(B') \subseteq E_\Phi\). Therefore, we see that \(\Psi \ll \Phi\) if and only if \(E_\Psi \subseteq E_\Phi\), equivalently, \(P'_\Psi \leq P'_\Phi\).
(2) By Lemma 25, two CP maps are mutually singular, \(\Psi \perp \Phi\), if and only if \(\Phi : \Psi = 0\). Since \(A'\) and \(B'\) commute, and \(A'+B'=I\), their operator parallel sum simplifies to \(A'(A'+B')^{-1}B' = A'B'\). Thus, \[(\Phi : \Psi)(x) = V^*(A'B')\pi(x)V = 0.\] By minimality, this is equivalent to \(A'B' = 0\). This implies \[\mathrm{ran}(B') \subseteq \mathrm{ker}(A') = E_\Phi^\perp,\] meaning the operators have mutually orthogonal supports. Therefore, \(\Psi \perp \Phi\) if and only if \(E_\Psi \perp E_\Phi\), equivalently, \(P'_\Psi P'_\Phi = 0\). ◻
Remark 26. In the above construction, the absolutely continuous part of \(\Psi\) admits a natural interpretation in terms of the shorted operator.
Indeed, since \(A', B' \in \pi(\mathcal{M})'\) are positive operators satisfying \(A'+B'=I\), they commute. Hence the operator limit \[\lim_{n\to\infty} (nA' : B')\] coincides with the shorted operator of \(B'\) to the subspace \(E_\Phi = \overline{\mathrm{ran}}(A')\).
The notion of shorted operator was originally introduced by Krein [18] and later developed by Anderson and Trapp [19], [20]. It is characterized as the maximal positive operator \(C \le B\) whose range is contained in \(E_\Phi\).
In the present commuting situation, this operator reduces to the simple compression. Thus, the Lebesgue decomposition of \(\Psi\) with respect to \(\Phi\) can be understood as the geometric operation of shorting the Arveson Radon–Nikodym derivative \(B'\) to the support subspace of \(\Phi\).
Example 10 (Recovery of Ando’s Lebesgue-type decomposition of bounded positive operators). We finally show that our framework recovers Ando’s decomposition. Let \(\mathcal{M}= \mathbb{C}\) and \(\mathcal{H}=L^2(\mathcal{N})\) be a standard Hilbert space of a von Neumann algebra \(\mathcal{N}\). A CP map from \(\mathbb{C}\) to \(\mathcal{N}\) is simply given by \(\Phi_A(z) = z A\) for some positive operator \(A \in \mathcal{N}^+\). Let \(\Phi_A\) and \(\Phi_B\) be CP maps corresponding to \(A\) and \(B\) in \(\mathcal{N}^+\), respectively.
Set \(C = A+B\). The minimal Stinespring dilation for \(\Gamma = \Phi_A + \Phi_B = \Phi_C\) can be realized explicitly on the subspace \(\mathcal{H}_\gamma = \overline{\mathrm{ran}}(C^{1/2}) \subseteq \mathcal{H}\). The representation is \(\pi(z) = z I_{\mathcal{H}_\gamma}\), and the bounded linear map \(V \colon \mathcal{H}\to \mathcal{H}_\gamma\) is given by \(V\xi = C^{1/2}\xi\).
The Arveson Radon-Nikodym derivatives \(A', B' \in \pi(\mathbb{C})' = \mathbb{B}(\mathcal{H}_\gamma)\) satisfying \(\Phi_A(z) = V^* A' \pi(z) V\) and \(\Phi_B(z) = V^* B' \pi(z) V\). Notice that \[C^{1/2}A'C^{1/2} = A, \quad C^{1/2}B'C^{1/2} = B.\] Recall that \[A' + B' = I_{\mathcal{H}_\gamma}.\] In particular, \(A'\) and \(B'\) commute. If \(A\) and \(B\) are invertible, then we have \[\begin{align} (\Phi_A : \Phi_B)(1) &= V^* (A' B') V \\ &= C^{1/2} (C^{-1/2} A C^{-1/2}) (C^{-1/2} B C^{-1/2}) C^{1/2} \\ &= A (A+B)^{-1} B=[B^{-1}(A+B)A^{-1}]^{-1} \\ &=(A^{-1}+B^{-1})^{-1}=A : B. \end{align}\] In general, by the standard arguments of the limits \(\lim_{\varepsilon\downarrow 0}(A+\varepsilon I)\) and \(\lim_{\varepsilon\downarrow 0}(B+\varepsilon I)\), we recover the classical parallel sum of positive operators \(A : B\).
Moreover, since \(n\Phi_A=\Phi_{nA}\), we have \[\begin{align} [\Phi_A]\Phi_B(1) &= \lim_{n\to\infty}(n\Phi_A : \Phi_B)(1) \\ &= \lim_{n\to\infty}(\Phi_{nA} : \Phi_B)(1) \\ &= \lim_{n\to\infty}(nA : B) \\ &= [A]B. \end{align}\]
Assuming \(\ker A = \{0\}\), Kosaki [37] considered the densely defined operator \(T=B^{1/2}A^{-1/2}\) with \(\mathrm{dom}(T)=\mathrm{ran}(A^{1/2})\). Since \(B^{1/2}\) is bounded, we have \(T^*=A^{-1/2}B^{1/2}\) and \[\mathrm{dom}(T^*)=\{\xi\in\mathcal{H}\mid B^{1/2}\xi\in \mathrm{ran}(A^{1/2})\}.\] Then \(T\) is closable, i.e., \(\mathrm{dom}(T^*)\) is dense, if and only if \(B\) is \(A\)-absolutely continuous ([37]). In general, if we do not assume non-singularity, then Kosaki [33] introduced unique contraction \(H\) satisfying \[A^{1/2}=H(A+B)^{1/2},\] where \(H\in \mathbb{B}(\mathcal{H})\) is defined by \(C^{1/2}\xi\mapsto A^{1/2}\xi\) and \(H=0\) on \((\mathrm{ran}(C^{1/2}))^\perp\). Then the operator \(B\) is \(A\)-absolutely continuous if and only if the operator \(H\) is injective ([33]).
Now we claim that \(A'=H^*H|_{\mathcal{H}_\gamma}\). Recall that \(C=A+B\in \mathbb{B}(\mathcal{H})\). Hence, \(A^{1/2}=HC^{1/2}\). Note that \[\mathopen{\langle}A\xi, \xi\mathclose{\rangle}=\mathopen{\langle}H^*HC^{1/2}\xi, C^{1/2}\xi\mathclose{\rangle},\] and \[\mathopen{\langle}A\xi, \xi\mathclose{\rangle}=\mathopen{\langle}A'C^{1/2}\xi, C^{1/2}\xi\mathclose{\rangle}.\] Since \(\mathrm{ran}(C^{1/2})\) is dense in \(\mathcal{H}_\gamma\), we conclude that \(A'=H^*H\) on \(\mathcal{H}_\gamma\).
Absolute Continuity: Our framework dictates that \(\Phi_B \ll \Phi_A\) if and only if \(E_{\Phi_B} \subseteq E_{\Phi_A}\), i.e., \(\overline{\mathrm{ran}}(B')\subseteq\overline{\mathrm{ran}}(A')\).
We first check that \[\overline{\mathrm{ran}}(B')\subseteq\overline{\mathrm{ran}}(A')\] if and only if \[\ker(A')=\{0\}.\] It is easy to see that \(\overline{\mathrm{ran}}(B')\subseteq\overline{\mathrm{ran}}(A')\) if and only if \(\ker(A')\subseteq\ker(B')\).
Suppose that \(\ker(A')\subseteq\ker(B')\). Let \(\eta\in\ker(A')\). Since \(\ker(A')\subseteq\ker(B')\), we have \(\eta\in\ker(B')\). Since \(A'+B'=I\), we obtain \[\eta=(A'+B')\eta=0.\] Hence, \(\ker(A')=\{0\}\). Conversely, if \(\ker(A')=\{0\}\), then \(\overline{\mathrm{ran}}(A')=\mathcal{H}_\gamma\), and so \(\overline{\mathrm{ran}}(B')\subseteq\overline{\mathrm{ran}}(A')\) holds.
Next we show that \(\ker(A')=\{0\}\) if and only if Kosaki’s \(A\)-absolutely continuity condition \(\ker(H)=\{0\}\). Indeed, suppose that \(\ker(H)=\{0\}\). If \(\eta\in \ker(A')\subseteq \mathcal{H}_\gamma\), then \[0=\mathopen{\langle}A'\eta, \eta\mathclose{\rangle} = \mathopen{\langle}H^*H\eta, \eta\mathclose{\rangle} = \|H\eta\|^2,\] which implies \(\eta=0\), because \(\ker(H)=\{0\}\). The converse is trivial, because \(\ker(H)=\ker(H^*H)\).
Singularity: In our framework, \(\Phi_A \perp \Phi_B\) if and only if \([\Phi_A]\Phi_B(1) = 0\), which is equivalent to \([A]B = 0\), i.e., \(B\) is \(A\)-singular.
Therefore, our Stinespring geometric framework provides a unified spatial perspective that not only recovers but conceptually explains Ando’s decomposition. In particular, it reveals that Ando’s intricate analytical density condition for absolute continuity is fundamentally equivalent to a simple closed subspace inclusion \(E_{\Phi_B} \subseteq E_{\Phi_A}\) at the level of the Arveson Radon-Nikodym derivatives.
The author used Gemini and ChatGPT for English language editing and mathematical information retrieval during the preparation of this manuscript.
The author was partially supported by JSPS KAKENHI Grant Number JP26K06843.↩︎