January 01, 1970
We study a quasi-product state associated with the one-dimensional hard-core Gibbs measure. After coding the model by the topological Markov chain, we construct the standard path \(AF\)-algebra of admissible hard-core words and show that the stationary Markov measure induces on it a faithful diagonal state in the sense of Evans. We then analyze the von Neumann algebra generated by the corresponding GNS representation. The resulting algebra is a hyperfinite factor, and its type is determined by the single parameter \(\kappa=q/p^2,\) where \(\begin{pmatrix}p&q\\ 1&0\end{pmatrix}\) is the transition matrix of the Markov chain. More precisely, the factor is of type \(\mathrm{II}_1\) when \(\kappa =1\), and of type \(\mathrm{III}_{\lambda}\) with \(\lambda=\min\{\kappa,\kappa^{-1}\}\) for \(\kappa\neq 1\). We also specify the centralizer and the weight flow for the resulting factor.
Quantum Markov states form a natural meeting point of operator algebras, probability, and statistical mechanics. Since the pioneering work of Accardi and his collaborators, noncommutative analogues of classical Markov chains have been studied from several points of view: the structure of states on quasi-local \(C^*\)-algebras, modular theory, and applications to concrete models from quantum statistical mechanics and quantum information theory; see, for example, [1]–[7]. One of the central problems in this circle of ideas is the determination of the type of the von Neumann algebra generated by the GNS representation of a given state.
A basic benchmark is provided by Powers’ classical construction [8]. He considered product states on UHF algebras, which can be interpreted as free quantum spin lattice systems with two local states at each site having probabilities \(p\) and \(q=1-p\). In the non-tracial case, after ordering the probabilities so that \(0<p\le q<1\), the corresponding factor is of type \(\mathrm{III}_{p/q}\). More extensive analyses of product states and their modular theory can be found in [9], [10]. From the physical point of view, these examples correspond to free systems, that is, systems without a genuine interaction between neighbouring sites.
Beyond the product case, two closely related lines of work are relevant for the present paper. On the one hand, Krieger’s work showed that factor-theoretic invariants of nonsingular dynamics, and in particular constructions based on Markov data, are closely tied to ratio sets and type \(\mathrm{III}\) phenomena [11], [12]. On the other hand, Evans introduced quasi-product measures on compact totally disconnected path spaces and the corresponding quasi-product states on associated unital AF-algebras and Cuntz–Krieger algebras [13], [14]. The present paper lies naturally at the intersection of these viewpoints: we construct a faithful quasi-product state arising from a stationary Markov measure on an admissible path space generated by a nearest-neighbor hard-core interaction and determine the type of the resulting factor by computing the ratio set of the tail equivalence relation.
This naturally leads to the question of what happens to general systems with interactions. According to Blekher’s commentary [15], one may expect that the presence of a nontrivial interaction should drive the associated factor toward type \(\mathrm{III}_1\). One of the main messages of the present paper is that this intuition may be too restrictive. It turns out that for the hard-core model under consideration the resulting factor is still of type \(\mathrm{III}_{\lambda}\) for an explicit \(\lambda\in(0,1)\), except for one exceptional case where it is of type \(\mathrm{II}_1\). In other words, a nontrivial interaction does not, by itself, force type \(\mathrm{III}_1\). Nevertheless, this area needs further studies for making more detailed conclusions.
It is worth noting that the case of a locally faithful Markov state has been treated in the literature. In Ref. [16] it was proved that non-homogeneous quantum Markov states are diagonalizable and showed that, in the translation-invariant or periodic locally faithful setting, the type of the generated von Neumann factor is determined by the spectrum of the fundamental two-site block of the associated Hamiltonian (see also [17], [18]). In particular, when the underlying stochastic matrix has no zero entries, so that the corresponding diagonal lifting gives a locally faithful Markov state, the factor-type problem falls within that framework. The hard-core model studied in the present paper lies outside this situation: its transition matrix contains a zero entry, and the diagonal state on the full UHF algebra is not locally faithful. It turns out that this fact affects the type of the resulting factor.
The model under consideration is the one-dimensional hard-core model on \({\mathbb{Z}}\). A configuration is a map \(\eta:{\mathbb{Z}}\to\{0,1\}\) satisfying the exclusion rule \[\eta(k)\eta(k+1)=0,\qquad k\in{\mathbb{Z}},\] so two neighbouring sites cannot be simultaneously occupied. For fixed fugacity, the Gibbs measure of this model is a stationary one-dimensional Markov measure; see [19]–[22]. We use the coding in which the vacant state is labelled by \(0\) and the occupied state by \(1\). The hard-core condition excludes the word \(11\), and the model is encoded by the topological Markov chain with states \(0\) and \(1\) and the adjacency matrix \(\begin{matrix}0\\ 1\end{matrix}\left[\begin{matrix}1&1\\ 1&0\end{matrix}\right]\). The corresponding stationary Markov measure \(\mu\) is determined by the transition matrix \(\begin{matrix}0\\ 1\end{matrix}\left[ \begin{matrix}p&q\\ 1&0\end{matrix}\right]\), with \(p,q\in(0,1)\), \(p+q=1\), and has the invariant distribution \(\big\{1/(1+q),q/(1+q)\big\}\).
If one starts from the full two-sided UHF algebra and defines the diagonal state coming from the Gibbs measure, the state is not faithful, because every cylinder containing the forbidden pattern \(11\) has zero weight. For the modular theory this is not the right ambient algebra; the correct object is instead the path AF-algebra built from admissible words. In this framework the forbidden pattern is excluded at the algebraic level, and the one-sided stationary Markov measure induces a faithful diagonal state. This places the construction naturally in the setting of quasi-product states introduced by Evans [13].
More precisely, for each \(n\geq 0\) we consider the finite-dimensional algebra \[{\mathcal{A}}_n=\operatornamewithlimits{\oplus}\limits_{j=0,1}{\mathcal{B}}\bigl(\ell^2(W_n(j))\bigr),\] where \(W_n(j)\) is the set of admissible words of length \(n+1\) ending at the symbol \(j\). Here \({\mathcal{B}}\bigl(\ell^2(W_n(j))\bigr)\) stands for the set of complex matrices wuth entries indexed by pairs of words from \(W_n(j)\). (Formal definitions are provided in Section 3.) The inductive limit \[{\mathcal{A}}=\varinjlim({\mathcal{A}}_n,\iota_n)\] is the path AF-algebra associated with the hard-core Bratteli diagram. The emerging stationary Markov measure induces a faithful state \(\omega\) on \({\mathcal{A}}\), diagonal in the natural matrix-unit basis and compatible with the inductive-limit structure.
Our main result gives a complete description of the type of the GNS factor generated by \(\omega\). The key quantity is the ratio \(q/p^2\), which measures the change in cylinder weights when a local block \(000\) is replaced by \(010\).
Theorem 1. The von Neumann algebra \({\mathcal{M}}=\pi_{\omega}({\mathcal{A}})''\), is a hyperfinite factor (HFF). If \(q=p^2\), then \({\mathcal{M}}\) is an HFF of type \(\mathrm{II}_1\). Otherwise \({\mathcal{M}}\) is of type \(\mathrm{III}_{\lambda}\) where \[\label{eqn1:1} \lambda=\min\left\{q/p^2,p^2/q\right\}\in(0,1).\qquad{(1)}\]
In particular, the hard-core model under consideration provides an example that does not lead to a factor of type \(\mathrm{III}_1\).
It is instructive to compare the above theorem with existing results on factor types arising in various models originated in physics. In this connection we have mentioned papers [16]–[18]. In addition, a non-trivial type-classification problem emerges for quantum Markov states on regular trees. In a number of models considered in Refs [23], [24], the resulting factors have type \(\mathrm{III}_{\lambda}\), with \(\lambda\in(0,1)\). From this perspective, the current work suggests that a full analysis of the types of von Neumann algebras arising from general Markov chains (classical or quantum), and more broadly from Markov processes on multidimensional lattices, is open.
The paper is organized as follows. In Section 2 we recall the modular-theoretic facts that are used later. Section 3 constructs the path AF-algebra of admissible hard-core words and the associated faithful quasi-product state. In Section 4 we prove that the generated von Neumann algebra is a hyperfinite factor and compute its Connes spectrum via the ratio set of the tail equivalence relation. In Section 5 we describe the centralizer of the state and the flow of weights of the resulting factor.
Let us recall facts from modular theory used in Section 4. Let \({\mathcal{M}}\) be a factor, \(\varphi\) a faithful normal state (FNS) on \({\mathcal{M}}\), and \(\sigma_t^{\varphi}\) the modular automorphism group associated with \(\varphi\). We write \(\Gamma(\sigma^\varphi)\) for the Connes spectrum of the modular automorphism group; see [9]. For a factor this invariant is independent of the faithful normal state \(\varphi\). The factor is of type \(\mathrm{III}_\lambda\), \(0<\lambda<1\), exactly when \(\Gamma(\sigma^\varphi)=(\log\lambda)\mathbb{Z},\) and it is of type \(\mathrm{III}_1\) exactly when \(\Gamma(\sigma^\varphi)=\mathbb{R}.\) cf. [25],[26]. On the other hand, if \({\mathcal{M}}\) admits a tracial FNS, then \({\mathcal{M}}\) is finite. In particular, an infinite-dimensional HFF is the HFF of type \(\mathrm{II}_1\); see [10], [25].
We write \(\Gamma(\sigma^\varphi)\) for the Connes spectrum of the modular automorphism group. For a factor this invariant is independent of the faithful normal state \(\varphi\). The factor is of type \(\mathrm{III}_\lambda\), \(0<\lambda<1\), exactly when \[\Gamma(\sigma^\varphi)=(\log\lambda)\mathbb{Z},\] and it is of type \(\mathrm{III}_1\) exactly when \[\Gamma(\sigma^\varphi)=\mathbb{R}.\]
We shall use the Feldman–Moore description of von Neumann algebras associated with measured equivalence relations; cf. [27], [28]. Let \(({\mathscr R},\mu)\) be a countable measured equivalence relation, let \(D\) be the Radon–Nikodým cocycle, and let \({\mathcal{F}}({\mathscr R},\mu)\) be the associated Feldman–Moore algebra. By [28], the modular automorphism group of the canonical state on \({\mathcal{F}}({\mathscr R},\mu)\) is implemented by the cocycle \(D^{it}\). Moreover, if \({\mathcal{F}}({\mathscr R},\mu)\) is a factor, then [28] identifies the Connes spectrum \(\Gamma({\mathcal{F}}({\mathscr R},\mu))\) with the nonzero part of the Krieger ratio set \({\rm K}({\mathscr R},\mu)\setminus\{0\}\): \[\label{KFM} \Gamma({\mathcal{F}}({\mathscr R},\mu))=\log\bigl({\rm K}({\mathscr R},\mu)\setminus\{0\}\bigr);\tag{1}\] see also [29]. This is the tool used in Section 4 to identify the type of the factor generated by the hard-core Gibbs measure.
Throughout the rest of the paper, symbol \(\Longrightarrow\) marks a logical implication and \(\Longleftrightarrow\) a logical equivalence.
It is convenient to pass from the two-sided hard-core Gibbs measure to its one-sided stationary Markov marginal. The admissibility condition excludes the word \(11\), so the natural algebraic object is the path \(AF\)-algebra associated with the hard-core adjacency matrix, rather than a quotient of the full UHF algebra.
Set \[\label{stoc} B=\left[\begin{matrix}1&1\\ 1&0\end{matrix}\right], \quad P=\left[\begin{matrix}p&q\\ 1&0\end{matrix} \right];\tag{2}\] as before, \(p,q\in(0,1)\) and \(p+q=1\). Next, define the collection \({\mathscr X}={\mathscr X}_B\) of one-sided admissible hard-core paths: \[\label{scrX}{\mathscr X}=\Big\{x=(x_0,x_1,x_2,\dots)\in\{0,1\}^{{\mathbb{Z}}_+};\; B (x_k,x_{k+1})=1\;\forall\;k\in{\mathbb{Z}}_+\Big\}.\tag{3}\] Here and below, \({\mathbb{Z}}_+=\{0,1,2,\ldots\}\), and \(B(i,j)\) stands for the \((i,j)\)th entry of matrix \(B\), where \(i,j\in\{0,1\}\). Set \({\mathscr X}\) is equipped with a standard Tykhonov topology (induced by that of \(\{0,1\}^{{\mathbb{Z}}_+}\)) in which it is a Polish space; we say that a set \({\mathscr A}\subset{\mathscr X}\) is measurable when \({\mathscr A}\) is a Borel set in this topology. For \(n\in\mathbb{Z}_+\), put \[\label{Wn}W_n=\Big\{\xi=(\xi_0,\xi_1,\dots,\xi_n)\in\{0,1\}^{n+1}:\; B(\xi_k,\xi_{k+1})=1,\;0\leq k<n\Big\},\tag{4}\] and, for \(j\in\{0,1\}\), \[\label{Wn40j41}W_n(j)=\Big\{\xi\in W_n:\;\xi_n=j\Big\}.\tag{5}\] As was noted, \(W_n\) is the collection of admissible words of length \(n+1\), while \(W_n(j)\) consists of those admissible words whose last symbol is \(j\). Next, we set \[\label{An} {\mathcal{A}}_n=\operatornamewithlimits{\oplus}\limits_{j=0,1}{\mathcal{B}}\bigl(\ell^2(W_n(j))\bigr).\tag{6}\]
For example, \(W_0=\{0,1\}\), \(W_1(0)=\{00,10\}\), \(W_1(1)=\{01\}\) \(\Longrightarrow\) \({\mathcal{A}}_1\cong M_2({\mathbb{C}})\oplus{\mathbb{C}}\), \({\mathcal{A}}_2\cong M_3({\mathbb{C}}) \oplus M_2({\mathbb{C}})\), and so on. This illustrates the basic point of the construction: all noncommutative matrix units are retained, but only inside blocks indexed by admissible words.
If \(\xi,\eta\in W_n(j)\) for the same \(j\), we denote by \(e^{(n)}_{\xi,\eta}\) the corresponding matrix units. They satisfy \[e^{(n)}_{\xi,\eta}e^{(n)}_{\zeta,\tau}=\delta_{\eta,\zeta}e^{(n)}_{\xi,\tau}, \qquad \bigl(e^{(n)}_{\xi,\eta}\bigr)^*=e^{(n)}_{\eta,\xi}.\] Here and below, \(\delta\) stands for the Kronecker symbol.
For \(n\in{\mathbb{Z}}_+\), define an embedding \(\iota_n:{\mathcal{A}}_n\to{\mathcal{A}}_{n+1}\) on matrix units by \[\label{embed} \iota_n\bigl(e^{(n)}_{\xi,\eta}\bigr) = \sum_{k:\,B(\xi_n,k)=1} e^{(n+1)}_{\xi k,\eta k},\tag{7}\] where \(\xi k=(\xi_0,\dots,\xi_n,k)\) and similarly for \(\eta k\). Since \(\xi_n=\eta_n\), the same admissible one-step extensions occur on both sides of 7 .
Proposition 1. For every \(n\in{\mathbb{Z}}_+\), the map \(\iota_n\) is a unital injective \(*\)-homomorphism. Hence the inductive limit \[\label{AFalg} {\mathcal{A}}=\varinjlim({\mathcal{A}}_n,\iota_n)\qquad{(2)}\] is an AF-algebra.
Proof. It is enough to check the assertion for the matrix units. Let \(\xi,\eta,\zeta,\tau\in W_n\) with \(\xi_n=\eta_n\) and \(\zeta_n=\tau_n\). Using 7 , we obtain \[\begin{align} \iota_n\bigl(e^{(n)}_{\xi,\eta}\bigr)\iota_n\bigl(e^{(n)}_{\zeta,\tau}\bigr) &= \sum_{k:\,B(\xi_n,k)=1}\sum_{l:\,B(\zeta_n,l)=1} e^{(n+1)}_{\xi k,\eta k}e^{(n+1)}_{\zeta l,\tau l}\\ &= \delta_{\eta,\zeta} \sum_{k:\,B(\xi_n,k)=1} e^{(n+1)}_{\xi k,\tau k} = \iota_n\bigl(e^{(n)}_{\xi,\eta}e^{(n)}_{\zeta,\tau}\bigr). \end{align}\] Similarly, \[\iota_n\bigl((e^{(n)}_{\xi,\eta})^*\bigr)=\iota_n(e^{(n)}_{\xi,\eta})^*.\] The image of the identity of \({\mathcal{A}}_n\) is the identity of \({\mathcal{A}}_{n+1}\) \(\Longrightarrow\) \(\iota_n\) is unital. Injectivity follows because each matrix block is embedded diagonally into a direct sum of matrix blocks. ◻
Given \(n\in{\mathbb{Z}}_+\), let \[{\mathcal{D}}_n=\operatorname{span}\{e^{(n)}_{\xi,\xi}:\;\xi\in W_n\}\subset {\mathcal{A}}_n\] stand for the diagonal subalgebra. Define a conditional expectation \({\mathcal{E}}_n:{\mathcal{A}}_n\to{\mathcal{D}}_n\) by \[\label{diagexp} {\mathcal{E}}_n\bigl(e^{(n)}_{\xi,\eta}\bigr)=\delta_{\xi,\eta}e^{(n)}_{\xi,\xi}.\tag{8}\] The family \(\{{\mathcal{E}}_n\}_{n\geq 0}\) is compatible with the embeddings \(\iota_n\), hence it induces a faithful conditional expectation \[\label{globalexp} {\mathcal{E}}:{\mathcal{A}}\to{\mathcal{D}}, \qquad {\mathcal{D}}=\varinjlim({\mathcal{D}}_n,\iota_n|_{{\mathcal{D}}_n}).\tag{9}\] Under the canonical identification \(e^{(n)}_{\xi,\xi}\longleftrightarrow \chi_{[\xi]}\), where \[\label{[xi]}[\xi]=\big\{x\in{\mathscr X}:\;x_0=\xi_0,\dots,x_n=\xi_n\big\},\tag{10}\] and \(\chi_{\mathscr A}\) stands for the indicator of a measurable set \({\mathscr A}\subset{\mathscr X}\), algebra \({\mathcal{D}}\) is naturally isomorphic to \(C({\mathscr X})\).
Let \(\pi=(\pi_0,\pi_1)\) be the invariant distribution of the matrix \(P\) \(\Longleftrightarrow\) \(\pi=\pi P\), where \(\pi_0=1/(1+q)\) and \(\pi_1=q/(1+q)\). Let \(\mu=\mu_P\) be the stationary Markov measure on \({\mathscr X}\) associated with \(P\) \(\Longleftrightarrow\) for every admissible word \(\xi=(\xi_0,\dots,\xi_n)\in W_n\), \[\label{cylinder} \mu([\xi])=\pi_{\xi_0}\prod_{k=0}^{n-1}P_{\xi_k\xi_{k+1}}.\tag{11}\] Here and below, \(P_{ij}\) denotes the \((i,j)\)-entry of \(P\). Since \(p,q\in(0,1)\), every admissible cylinder has strictly positive \(\mu\)-measure.
For each \(n\in{\mathbb{Z}}_+\) define a state \(\omega_{P,n}\) on \({\mathcal{A}}_n\) by \[\label{localstate} \omega_{P,n}\bigl(e^{(n)}_{\xi,\eta}\bigr) = \delta_{\xi,\eta}\mu([\xi]), \qquad \xi,\eta\in W_n,\;\xi_n=\eta_n.\tag{12}\]
Proposition 2. The family \(\{\omega_{P,n}\}_{n\geq 0}\) is compatible with the embeddings \(\iota_n\). Consequently, there exists a unique state \(\omega=\omega_P\) on \({\mathcal{A}}\) such that \[\omega|_{{\mathcal{A}}_n}=\omega_{P,n} \qquad (n\in{\mathbb{Z}}_+).\] Moreover, \[\label{diagstate} \omega(x)=\mu\bigl({\mathcal{E}}(x)\bigr), \qquad x\in {\mathcal{A}},\qquad{(3)}\] and \(\omega\) is an FNS.
Proof. It is enough to check compatibility on the matrix units. Let \(\xi,\eta\in W_n\) with \(\xi_n=\eta_n\). By 7 and 12 , \[\begin{align} \omega_{P,n+1}\bigl(\iota_n(e^{(n)}_{\xi,\eta})\bigr) &= \sum_{k:\,B(\xi_n,k)=1} \omega_{P,n+1}\bigl(e^{(n+1)}_{\xi k,\eta k}\bigr)= \delta_{\xi,\eta} \sum_{k:\,B(\xi_n,k)=1} \mu([\xi k])\\ &= \delta_{\xi,\eta}\mu([\xi])\sum_{k:\,B(\xi_n,k)=1}P_{\xi_n k}= \delta_{\xi,\eta}\mu([\xi]) = \omega_{P,n}\bigl(e^{(n)}_{\xi,\eta}\bigr). \end{align}\] Therefore the inductive-limit state \(\omega\) is well defined. Formula ?? follows from 8 and 12 . Since \({\mathcal{E}}\) is faithful and \(\mu([\xi])>0\) for every admissible cylinder, the state \(\omega\) is faithful. ◻
The diagonal form of \(\omega\) on the canonical matrix units shows that, in the terminology of Evans [13], it is a faithful quasi-product state on the path AF-algebra \({\mathcal{A}}\).
For later use, let \(\mathop{\mathrm{Tr}}_n\) denote the canonical unnormalized trace on \({\mathcal{A}}_n\), characterized by \[\mathop{\mathrm{Tr}}_n\bigl(e^{(n)}_{\xi,\eta}\bigr)=\delta_{\xi,\eta}.\] Then the restriction of \(\omega\) to \({\mathcal{A}}_n\) is given by \[\label{density} \omega(x)=\mathop{\mathrm{Tr}}_n(\rho_n x), \qquad x\in {\mathcal{A}}_n,\tag{13}\] where \[\label{rho} \rho_n= \sum_{\xi\in W_n}\mu([\xi])e^{(n)}_{\xi,\xi} = \sum_{\xi\in W_n} \left( \pi_{\xi_0}\prod_{k=0}^{n-1}P_{\xi_k\xi_{k+1}} \right)e^{(n)}_{\xi,\xi}\tag{14}\] is the local density matrix. Since all diagonal coefficients are strictly positive, the local Hamiltonian \(H_n=-\log \rho_n\) is well defined and has the form \[\label{Hnnew} H_n= -\sum_{\xi\in W_n} \left( \log\pi_{\xi_0}+\sum_{k=0}^{n-1}\log P_{\xi_k\xi_{k+1}} \right)e^{(n)}_{\xi,\xi}.\tag{15}\] In particular, the stationary boundary term \(\log\pi_{\xi_0}\) is part of the finite-volume Hamiltonian.
Throughout Sections 4 and 5, we set \(\kappa=q/p^2\), \(\lambda=\min [\kappa, \kappa^{-1}]\) and let \({\mathcal{M}}:=\pi_{\omega} ({\mathcal{A}})''\) be the von Neumann algebra generated by the GNS representation of state \(\omega\) constructed in Proposition 2; cf. ?? . Since \(\omega\) is faithful, we identify \({\mathcal{A}}\) with its image in \({\mathcal{M}}\) and regard \(\omega\) as an FNS of \({\mathcal{M}}\).
Given \(n\in{\mathbb{Z}}_+\) and \(j\in\{0,1\}\), put \[\label{centproj4} {\rm p}_j^{(n)}=\sum_{\xi\in W_n(j)} e^{(n)}_{\xi,\xi}.\tag{16}\] Then \({\mathcal{A}}_n\) has the center \(Z({\mathcal{A}}_n)={\mathbb{C}}\,{\rm p}_0^{(n)}\oplus {\mathbb{C}}\,{\rm p}_1^{(n)}\).
Proposition 3. Let \(\left\{\sigma_t^{\omega}\right\}\) be the modular group for state \(\omega\). Then, for every \(n\geq 0\) and every \(\xi,\eta\in W_n\) with \(\xi_n=\eta_n\), \[\label{modunits4} \sigma_t^{\omega}\bigl(e^{(n)}_{\xi,\eta}\bigr) =\left(\frac{\mu([\xi])}{\mu([\eta])}\right)^{it}e^{(n)}_{\xi,\eta}, \qquad t\in{\mathbb{R}}.\qquad{(4)}\] In particular, each local matrix unit is an entire analytic element for \(\left\{\sigma_t^{\omega}\right\}\).
Proof. Let \((H,\pi,\Omega)\) be the GNS triple of \(({\mathcal{A}},\omega)\) We identify \({\mathcal{A}}\) with its image \(\pi({\mathcal{A}})\) in the algebra \({\mathcal{B}}(H)\) of bounded operators in \(H\) and use the convention \(\langle x\Omega,y\Omega\rangle=\omega(x^*y)\), where \(x,y\in {\mathcal{A}}\).
Let \[S_0(x\Omega)=x^*\Omega, \qquad x\in {\mathcal{A}},\] be the Tomita operator on the dense domain \({\mathcal{A}}\Omega\), and let \(S=\overline{S_0}\) be its closure. Then \(\Delta=S^*S\) is the modular operator associated with \(\omega\).
Fix \(n\in{\mathbb{Z}}_+\) and \(\xi,\eta\in W_n\) with \(\xi_n=\eta_n\). Set \(v_{\xi,\eta}:=e^{(n)}_{\xi,\eta}\Omega\) \(\Longrightarrow\) \(v_{\xi,\eta}\in\operatorname{Dom}(S)\) and \(Sv_{\xi,\eta}= \bigl(e^{(n)}_{\xi,\eta}\bigr)^*\Omega= e^{(n)}_{\eta,\xi}\Omega=v_{\eta,\xi}\).
We claim that \[S^*v_{\xi,\eta} = \frac{\mu([\eta])}{\mu([\xi])}\,v_{\eta,\xi}.\] To prove this, let \(m\geq n\), put \(j:=\xi_n=\eta_n\), and denote by \(\operatorname{Ext}_{m-n}(j)\) the set of admissible continuations \(\tau=(\tau_1,\dots,\tau_{m-n})\) such that \(B(j,\tau_1)=1\) and \(B(\tau_r,\tau_{r+1})=1\) for \(1\le r<m-n\). For \(\tau\in \operatorname{Ext}_{m-n}(j)\), write \(\xi\tau\) and \(\eta\tau\) for the concatenated words in \(W_m\). By iterating 7 , we have \[e^{(n)}_{\xi,\eta} = \sum_{\tau\in \operatorname{Ext}_{m-n}(j)} e^{(m)}_{\xi\tau,\eta\tau}, \qquad e^{(n)}_{\eta,\xi} = \sum_{\tau\in \operatorname{Ext}_{m-n}(j)} e^{(m)}_{\eta\tau,\xi\tau}.\]
Now let \(\alpha,\beta,\gamma,\tau\in W_m\) with \(\alpha_m=\beta_m\) and \(\gamma_m=\tau_m\). By 12 , \[\langle e^{(m)}_{\alpha,\beta}\Omega,e^{(m)}_{\gamma,\tau}\Omega\rangle = \delta_{\alpha,\gamma}\delta_{\beta,\tau}\,\mu([\beta]).\] Therefore \[\begin{align} \bigl\langle S_0(e^{(m)}_{\alpha,\beta}\Omega),\,v_{\xi,\eta}\bigr\rangle &= \bigl\langle e^{(m)}_{\beta,\alpha}\Omega,e^{(n)}_{\xi,\eta}\Omega\bigr\rangle = \sum_{\tau\in \operatorname{Ext}_{m-n}(j)} \bigl\langle e^{(m)}_{\beta,\alpha}\Omega,e^{(m)}_{\xi\tau,\eta\tau}\Omega\bigr\rangle\\ &= \sum_{\tau\in \operatorname{Ext}_{m-n}(j)} \delta_{\beta,\xi\tau}\delta_{\alpha,\eta\tau}\,\mu([\alpha]). \end{align}\] On the other hand, \[\begin{align} \left\langle e^{(m)}_{\alpha,\beta}\Omega, \frac{\mu([\eta])}{\mu([\xi])}\,v_{\eta,\xi}\right\rangle &= \frac{\mu([\eta])}{\mu([\xi])} \bigl\langle e^{(m)}_{\alpha,\beta}\Omega,e^{(n)}_{\eta,\xi}\Omega\bigr\rangle = \frac{\mu([\eta])}{\mu([\xi])} \sum_{\tau\in \operatorname{Ext}_{m-n}(j)} \bigl\langle e^{(m)}_{\alpha,\beta}\Omega,e^{(m)}_{\eta\tau,\xi\tau}\Omega\bigr\rangle\\ &=\frac{\mu([\eta])}{\mu([\xi])} \sum_{\tau\in \operatorname{Ext}_{m-n}(j)} \delta_{\alpha,\eta\tau}\delta_{\beta,\xi\tau}\,\mu([\beta]). \end{align}\] Since \(\xi_n=\eta_n=j\), appending the same continuation \(\tau\) multiplies \(\mu([\xi])\) and \(\mu([\eta])\) by the same factor, so \[\frac{\mu([\eta\tau])}{\mu([\xi\tau])} = \frac{\mu([\eta])}{\mu([\xi])}.\] Hence the last two displayed expressions are equal. Since \(\operatornamewithlimits{\cup}\limits_{m\geq n}{\mathcal{A}}_m\Omega\) is dense in \(H\), the claim follows.
Consequently, \[\Delta v_{\xi,\eta} = S^*Sv_{\xi,\eta} = S^*v_{\eta,\xi} = \frac{\mu([\xi])}{\mu([\eta])}\,v_{\xi,\eta}.\] Thus \(v_{\xi,\eta}\) is an eigenvector of \(\Delta\), and for every \(t\in{\mathbb{R}}\), \[\Delta^{it}e^{(n)}_{\xi,\eta}\Omega = \left(\frac{\mu([\xi])}{\mu([\eta])}\right)^{it} e^{(n)}_{\xi,\eta}\Omega.\] Since \(\Delta^{it}\Omega=\Omega\), we obtain \[\sigma_t^{\omega}\bigl(e^{(n)}_{\xi,\eta}\bigr)\Omega = \Delta^{it}e^{(n)}_{\xi,\eta}\Delta^{-it}\Omega = \left(\frac{\mu([\xi])}{\mu([\eta])}\right)^{it} e^{(n)}_{\xi,\eta}\Omega.\] Because \(\Omega\) is separating for \({\mathcal{M}}\), ?? follows.
Finally, \(\mu([\xi])/\mu([\eta])>0\), so the map \[z\longmapsto \exp\!\left( iz\log\frac{\mu([\xi])}{\mu([\eta])} \right)e^{(n)}_{\xi,\eta}, \qquad z\in{\mathbb{C}},\] is entire and restricts on the real axis to \(t\mapsto \sigma_t^{\omega}(e^{(n)}_{\xi,\eta})\). Therefore \(e^{(n)}_{\xi,\eta}\) is an entire analytic element for \(\sigma^{\omega}\). ◻
Proposition 4. The von Neumann algebra \({\mathcal{M}}\) is an HFF.
Proof. AF algebra \({\mathcal{A}}\) coincides with the norm-closure \({\mathcal{A}}=\overline{\operatornamewithlimits{\cup}\limits_{n\geq 0}{\mathcal{A}}_n}^{\|\cdot\|}\) and each \({\mathcal{A}}_n\) is finite-dimensional. Therefore, \[{\mathcal{M}}=\pi_{\omega}({\mathcal{A}})''=\left(\operatornamewithlimits{\cup}\limits_{n\geq 0}\pi_{\omega}({\mathcal{A}}_n)\right)''.\] Thus, \({\mathcal{M}}\) is generated by an increasing sequence of finite-dimensional \(*\)-subalgebras, so \({\mathcal{M}}\) is hyperfinite.
It remains to prove that the center \(Z({\mathcal{M}})={\mathbb{C}}\mathbf{1}\). For every \(n\in{\mathbb{Z}}_+\), Proposition 3 shows that \(\sigma_t^{\omega}({\mathcal{A}}_n)={\mathcal{A}}_n\), \(t\in{\mathbb{R}}\).
Since \({\mathcal{A}}_n\) is finite-dimensional, it is weakly closed in \({\mathcal{M}}\). Therefore, by Takesaki’s theorem [10], there exists a unique \(\omega\)-preserving normal conditional expectation \(F_n:{\mathcal{M}}\to {\mathcal{A}}_n\).
Let \(z\in Z({\mathcal{M}})\) and set \[z_n:=F_n(z)\in {\mathcal{A}}_n.\] Because \(F_n\) is \({\mathcal{A}}_n\)-bimodular, for every \(x\in {\mathcal{A}}_n\) we have \(xz_n=F_n(xz)=F_n(zx)=z_nx\) \(\Longrightarrow\) \(z_n\in Z({\mathcal{A}}_n)\). By 16 , there exist scalars \(a_n,b_n\in{\mathbb{C}}\) such that \[\label{znform4} z_n=a_n{\rm p}_0^{(n)}+b_n{\rm p}_1^{(n)}.\tag{17}\]
We next compute the expectations of projections \({\rm p}_0^{(n+1)}\) and \({\rm p}_1^{(n+1)}\). Since \({\rm p}_j^{(n+1)}\) commutes with \({\mathcal{A}}_n\), the element \(F_n({\rm p}_j^{(n+1)})\) belongs to \(Z({\mathcal{A}}_n)\). Thus there exist \(\alpha,\beta,\gamma,\delta\in{\mathbb{C}}\) such that \[F_n\bigl({\rm p}_0^{(n+1)}\bigr)=\alpha\,{\rm p}_0^{(n)}+\beta\,{\rm p}_1^{(n)}, \qquad F_n\bigl({\rm p}_1^{(n+1)}\bigr)=\gamma\,{\rm p}_0^{(n)}+\delta\,{\rm p}_1^{(n)}.\]
Let \(\xi\in W_n(0)\), then \[\begin{align} \alpha\,\mu([\xi]) &= \omega\Bigl(e^{(n)}_{\xi,\xi}F_n\bigl({\rm p}_0^{(n+1)}\bigr)\Bigr)= \omega\Bigl(e^{(n)}_{\xi,\xi}{\rm p}_0^{(n+1)}\Bigr)= \omega\bigl(e^{(n+1)}_{\xi 0,\xi 0}\bigr)\\ &= \mu([\xi 0])= p\,\mu([\xi]). \end{align}\] Hence \(\alpha=p\).
Similarly, if \(\xi\in W_n(1)\), then \[\begin{align} \beta\,\mu([\xi]) &= \omega\Bigl(e^{(n)}_{\xi,\xi}F_n\bigl({\rm p}_0^{(n+1)}\bigr)\Bigr)= \omega\Bigl(e^{(n)}_{\xi,\xi}{\rm p}_0^{(n+1)}\Bigr)= \omega\bigl(e^{(n+1)}_{\xi 0,\xi 0}\bigr)\\ &= \mu([\xi 0])= \mu([\xi]), \end{align}\] because \(P_{10}=1\). Therefore, \(\beta=1\) \(\Longrightarrow\) \(F_n\bigl({\rm p}_0^{(n+1)}\bigr)=p\,{\rm p}_0^{(n)}+{\rm p}_1^{(n)}.\)
Next, let \(\xi\in W_n(0)\). Then \[\begin{align} \gamma\,\mu([\xi])= \omega\Bigl(e^{(n)}_{\xi,\xi}F_n\bigl({\rm p}_1^{(n+1)}\bigr)\Bigr)= \omega\bigl(e^{(n+1)}_{\xi 1,\xi 1}\bigr) =\mu([\xi 1])=q\,\mu([\xi]). \end{align}\] Hence \(\gamma=q\). If \(\xi\in W_n(1)\), then there is no admissible extension of \(\xi\) by \(1\), so \(e^{(n)}_{\xi,\xi}{\rm p}_1^{(n+1)}=0\) \(\Longrightarrow\) \(\delta=0\). Therefore \[\label{Fncenters4} F_n\bigl({\rm p}_0^{(n+1)}\bigr)=p\,{\rm p}_0^{(n)}+{\rm p}_1^{(n)}, \qquad F_n\bigl({\rm p}_1^{(n+1)}\bigr)=q\,{\rm p}_0^{(n)}.\tag{18}\]
Since \({\mathcal{A}}_n\subset {\mathcal{A}}_{n+1}\), the map \(F_n\circ F_{n+1}\) is again a \(\omega\)-preserving conditional expectation from \({\mathcal{M}}\) onto \({\mathcal{A}}_n\). By uniqueness, we have \(F_n\circ F_{n+1}=F_n\). Applying this identity to \(z\) and using 17 and 18 , we obtain \[z_n = F_n(z_{n+1}) = a_{n+1}F_n\bigl({\rm p}_0^{(n+1)}\bigr)+b_{n+1}F_n\bigl({\rm p}_1^{(n+1)}\bigr),\] hence \[\label{recurrenceab4} a_n=pa_{n+1}+qb_{n+1}, \qquad b_n=a_{n+1}.\tag{19}\]
Set \(d_n:=a_n-b_n\). Then 19 yields \[d_n = (pa_{n+1}+qb_{n+1})-a_{n+1} = -q(a_{n+1}-b_{n+1}) = -q\,d_{n+1}.\] Iterating, we get \[d_n=(-q)^m d_{n+m},\;\;and\;\;\|z_{n+m}\|=\|F_{n+m}(z)\|\le \|z\|, \;\; m\geq 1.\] Since \[z_{n+m}=a_{n+m}{\rm p}_0^{(n+m)}+b_{n+m}{\rm p}_1^{(n+m)}\] with \({\rm p}_0^{(n+m)}\) and \({\rm p}_1^{(n+m)}\) orthogonal projections, we have \[\|z_{n+m}\|=\max\{|a_{n+m}|,|b_{n+m}|\}.\] Therefore, \[|d_{n+m}|\le |a_{n+m}|+|b_{n+m}|\le 2\|z\|\quad\Longrightarrow\quad |d_n|\le 2\|z\|\,q^m\quad (m\geq 1).\] Letting \(m\to\infty\) gives \(d_n=0\). Hence \(a_n=b_n\) for all \(n\in{\mathbb{Z}}_+\) \(\Longrightarrow\) \(z_n\in{\mathbb{C}}\mathbf{1}\) for all \(n\in{\mathbb{Z}}_+\).
Let \((H,\pi,\Omega)\) be the GNS triple of \(({\mathcal{A}},\omega)\); as before, we identify \({\mathcal{A}}\subset{\mathcal{M}}\subset{\mathcal{B}}(H)\). Since \(F_n\) is \(\omega\)-preserving, the map \(P_n(x\Omega):=F_n(x)\Omega\), \(x\in {\mathcal{M}}\), extends to an orthogonal projection of \(H=L^2({\mathcal{M}},\omega)\) onto \(\overline{{\mathcal{A}}_n\Omega}\). Indeed, if \(x\in{\mathcal{M}}\) and \(a\in{\mathcal{A}}_n\), then \[\begin{align} \langle (x-F_n(x))\Omega,a\Omega\rangle &= \omega\bigl((x-F_n(x))^*a\bigr) =\omega(x^*a)-\omega\bigl(F_n(x)^*a\bigr)\\ &=\omega(x^*a)-\omega\bigl(F_n(x^*a)\bigr)=0. \end{align}\] The subspaces \(\overline{{\mathcal{A}}_n\Omega}\) are increasing, and \(\overline{\operatornamewithlimits{\cup}\limits_{n\geq 0}{\mathcal{A}}_n\Omega}=H\) because \(\operatornamewithlimits{\cup}\limits_{n\geq 0}{\mathcal{A}}_n\) is norm-dense in \({\mathcal{A}}\) and \(\Omega\) is cyclic. Hence \(P_n\to I_H\) strongly, and therefore \(\|F_n(z)\Omega-z\Omega\|\longrightarrow 0\).
But each \(F_n(z)=z_n\) is scalar, so every vector \(F_n(z)\Omega\) lies in the one-dimensional closed subspace \({\mathbb{C}}\Omega\). Therefore \(z\Omega\in{\mathbb{C}}\Omega\), say \(z\Omega=c\Omega\). Since \(\Omega\) is separating for \({\mathcal{M}}\), we conclude that \(z=c\mathbf{1}.\) Thus \(Z({\mathcal{M}})={\mathbb{C}}\mathbf{1}\), and \({\mathcal{M}}\) is a factor. ◻
We now compute the Connes spectrum of the modular group \(\{\sigma_t^{\omega}\}\). Let \[{\mathscr R}= \{(x,y)\in{\mathscr X}\times{\mathscr X}:\;\exists N \text{ such that } x_k=y_k \text{ for all } k\geq N\}\] be the tail equivalence relation on \({\mathscr X}\). Under the canonical identification of \({\mathcal{A}}\) with the AF groupoid algebra of \({\mathscr R}\), the pair \(({\mathcal{M}},\omega)\) is the Feldman–Moore algebra of the measured equivalence relation \(({\mathscr R},\mu)\); see [27], [28]. If \((x,y)\in{\mathscr R}\) and \(N\) is such that \(x_k=y_k\) for all \(k\geq N\), then the Radon–Nikodým cocycle is given by \[\label{cocycle4} D(x,y)=\frac{\pi_{x_0}}{\pi_{y_0}} \prod_{k=0}^{N-1}\frac{P_{x_kx_{k+1}}}{P_{y_ky_{k+1}}}.\tag{20}\] Let \({\rm K}({\mathscr R},\mu)\) denote the Krieger ratio set of this cocycle. By [28], the modular automorphism group of the canonical state on the Feldman–Moore algebra is implemented by \(D^{it}\), and \(S({\mathcal{M}})\setminus\{0\}={\rm K}({\mathscr R},\mu)\setminus\{0\}\); see also [27] and [29]. Since \({\mathcal{M}}\) is a factor, \(S({\mathcal{M}})\setminus\{0\}=\exp\Gamma(\sigma^{\omega})\). Therefore \[\label{Connesratio4} \Gamma\bigl(\sigma^{\omega}\bigr) = \log\bigl({\rm K}({\mathscr R},\mu)\setminus\{0\}\bigr).\tag{21}\]
Let \[{\mathscr X}_0=[0]=\{x\in{\mathscr X}:\;x_0=0\}\] and let \({\mathscr R}_0:={\mathscr R}|_{{\mathscr X}_0}\) be the restricted tail relation. Since \(\mu({\mathscr X}_0)=\pi_0>0\) and every \({\mathscr R}\)-class meets \({\mathscr X}_0\), the standard restriction property of the ratio set gives \[\label{restrictionratio4} {\rm K}({\mathscr R},\mu)={\rm K}({\mathscr R}_0,\mu|_{{\mathscr X}_0}).\tag{22}\]
Lemma 5. For an admissible word \(\xi=(0,\xi_1,\dots,\xi_n)\) starting at \(0\), define \[m(\xi):=\#\{0\le k<n:\;\xi_k=0,\;\xi_{k+1}=1\}.\] Suppose \(\xi\) and \(\eta\) are admissible words of the same length, both start at \(0\), and end at the same symbol. Then \(\mu([\xi])/\mu([\eta])=\kappa^{m(\xi)-m(\eta)}\). In particular, \(\mu([010])/\mu([000])=\kappa\).
Proof. Assume first that \(\xi\) ends at \(0\). Since the only admissible transitions are \(00\), \(01\), and \(10\), the number of \(10\) transitions in \(\xi\) equals \(m(\xi)\), and therefore the number of \(00\) transitions is \(n-2m(\xi)\). Hence, \(\mu([\xi])=\pi_0\,p^{n-2m(\xi)}q^{m(\xi)}=\pi_0\,p^n\kappa^{m(\xi)}\).
If \(\xi\) ends at \(1\), then the number of \(10\) transitions is \(m(\xi)-1\), so the number of \(00\) transitions is \(n+1-2m(\xi)\). Therefore, \(\mu([\xi])=\pi_0\,p^{n+1-2m(\xi)}q^{m(\xi)}= \pi_0\,p^{n+1}\kappa^{m(\xi)}\), and a similar formula holds for \(\eta\). Since \(\xi\) and \(\eta\) have the same length and the same terminal symbol, the common prefactor cancels, and we obtain the stated ratio formula. Taking \(\xi=010\) and \(\eta=000\) gives the last claim. ◻
Lemma 6. Let \({\mathscr A}\subset {\mathscr X}_0\) be measurable with \(\mu({\mathscr A})>0\), and let \(\delta>0\). Then there exists an admissible word \(u\) which starts and ends at \(0\) such that \(\mu({\mathscr A}\cap [u])>(1-\delta)\mu([u])\).
Proof. Let \({\mathfrak D}_n\) be the finite \(\sigma\)-algebra generated by cylinders of length \(n+1\) inside \({\mathscr X}_0\). By martingale convergence, \(\mathbb{E}(\chi_{{\mathscr A}}\mid {\mathfrak D}_n)(x)\longrightarrow \chi_{{\mathscr A}}(x)\) for \(\mu\)-almost every \(x\in {\mathscr X}_0\). Choose \(x\in {\mathscr A}\) such that the above convergence holds. Every admissible path contains infinitely many symbols \(0\), so there exists \(n\geq 0\) such that \(x_n=0\) and \(\mathbb{E}(\chi_{{\mathscr A}}\mid {\mathfrak D}_n)(x)>1-\delta\). If \(u=(x_0,x_1,\dots,x_n)\) is the corresponding prefix, then \(u\) starts and ends at \(0\), and the conditional expectation identity yields \(\mu({\mathscr A}\cap [u])>(1-\delta)\mu([u])\). ◻
Proposition 7. One has \[\label{ratiosetformula4} {\rm K}({\mathscr R},\mu)\setminus\{0\}=\kappa^{{\mathbb{Z}}}.\qquad{(5)}\] Equivalently, \[\label{Connesspecformula4} \Gamma\bigl(\sigma^{\omega}\bigr)=(\log\kappa){\mathbb{Z}}.\qquad{(6)}\]
Proof. Let \((x,y)\in{\mathscr R}_0\). Choose \(N\) such that \(x_k=y_k\) for all \(k\geq N\), and set \(\xi=(x_0,\dots,x_N)\), \(\eta=(y_0,\dots,y_N)\). Then \(\xi\) and \(\eta\) have the same length, both start at \(0\), and end at the same symbol. Moreover, by 20 , \(D(x,y)=\mu([\xi])/\mu([\eta])\).
Lemma 5 therefore implies that every cocycle value on \({\mathscr R}_0\) belongs to \(\kappa^{{\mathbb{Z}}}\). Hence \[\label{ratioUpper4} {\rm K}({\mathscr R}_0,\mu|_{{\mathscr X}_0})\setminus\{0\}\subset\kappa^{{\mathbb{Z}}}.\tag{23}\]
Conversely, we show that \(\kappa\) itself belongs to the ratio set. Let \({\mathscr A}\subset {\mathscr X}_0\) be measurable with \(\mu({\mathscr A})>0\). Choose \(\delta>0\) so small that \[\label{deltachoice4} p^2-(1+\kappa^{-1})\delta>0.\tag{24}\] By Lemma 6, there exists an admissible word \(u\) which starts and ends at \(0\) such that \[\label{densitycyl4} \mu({\mathscr A}\cap [u])>(1-\delta)\mu([u]).\tag{25}\] Define a partial isomorphism \(T_u:[u00]\to [u10]\), with \(T_u(u00z)=u10z\). The graph of \(T_u\) is contained in \({\mathscr R}_0\), and the Radon–Nikodým derivative is constant: \[\label{constantder4} \frac{d(\mu\circ T_u)}{d\mu}(x)=\kappa, \qquad x\in [u00].\tag{26}\] Indeed, \(\mu([u00])=p^2\mu([u])\), and \(\mu([u10])=q\mu([u])=\kappa\,\mu([u00])\).
Now put \[{\mathscr B}:={\mathscr A}\cap [u00]\cap T_u^{-1}({\mathscr A}\cap [u10]).\] Using 24 , 25 and 26 , we obtain \[\begin{align} \mu({\mathscr B}) &\geq \mu([u00]) -\mu([u00]\setminus {\mathscr A}) -\mu\bigl([u00]\setminus T_u^{-1}({\mathscr A}\cap [u10])\bigr)\\ &\geq p^2\mu([u]) -\delta\mu([u]) -\kappa^{-1}\delta\mu([u])>0. \end{align}\] Therefore, \(T_u\) sends a positive-measure subset of \({\mathscr A}\) into \({\mathscr A}\) and has
the derivative \(\kappa\) there. Thus, \(\kappa\in{\rm K}({\mathscr R}_0,\mu|_{{\mathscr X}_0})\).
By Proposition 4, \({\mathcal{M}}\) is a factor. Hence, relation \({\mathscr R}\) is ergodic, and so
is \({\mathscr R}_0\); see [27], [28]. For an ergodic measured equivalence relation the nonzero part of the ratio set is a closed subgroup of \((0,\infty)\). Therefore, \(\kappa^{{\mathbb{Z}}}\subset {\rm K}({\mathscr R}_0,\mu|_{{\mathscr X}_0})\setminus\{0\}\). Together with 23 and 22 , this proves ?? . Finally, ?? follows from 21 by taking logarithms. ◻
cm
Summarizing, we obtain the result announced in Theorem 1: cm
Theorem 8. (i) If \(\kappa\neq 1\), then \({\mathcal{M}}\) is an HFF of type \(\mathrm{III}_{\lambda}\).
(ii) If \(\kappa=1\), then \({\mathcal{M}}\) is an HFF of type \(\mathrm{II}_1\).
Proof. (i) Assume first that \(\kappa\neq 1\). Proposition 7 yields that \(\Gamma\bigl(\sigma^{\omega}\bigr)=(\log\kappa){\mathbb{Z}}=(\log\lambda){\mathbb{Z}}\). Since \({\mathcal{M}}\) is a factor, the classification recalled in Section 2 implies that \({\mathcal{M}}\) is of type \(\mathrm{III}_{\lambda}\).
(ii). Now let \(\kappa=1\) \(\Longleftrightarrow\) \(q=p^2\). In this case, for every admissible word \(\xi=(\xi_0,\dots,\xi_n)\) ending at \(j\in\{0,1\}\) one has \(\mu([\xi])=\pi_0\,p^{n+\delta_{j,1}+\delta_{\xi_0,1}}\). Hence, if \((x,y)\in {\mathscr R}\) and \(x_k=y_k\) for all \(k\geq N\), then \[\label{coboundary4} D(x,y)=p^{\delta_{x_0,1}-\delta_{y_0,1}}=\frac{h(x)}{h(y)}, \qquad h(x):=p^{\delta_{x_0,1}}.\tag{27}\] Thus the Radon–Nikodým cocycle is a coboundary. Define a finite measure \(\nu\) on \({\mathscr X}\) by \({\rm d}\nu=h^{-1}\,{\rm d}\mu\). Equation 27 implies that \(\nu\) is \({\mathscr R}\)-invariant \(\Longrightarrow\) the normal state on \({\mathcal{M}}\) induced by \(\nu\) is a tracial FNS \(\Longrightarrow\) \({\mathcal{M}}\) is a finite factor. Since \({\mathcal{M}}\) is infinite-dimensional and hyperfinite, it is the hyperfinite factor of type \(\mathrm{II}_1\). ◻
(1) If \(p=q=1/2\), then \(\kappa=2\) \(\Longrightarrow\) \({\mathcal{M}}\) is of type \(\mathrm{III}_{1/2}\).
(2) If \(p=(\sqrt5-1)/2\), then \(\kappa =1\) \(\Longrightarrow\) \({\mathcal{M}}\) is of type \(\mathrm{II}_1\).
(3) If \(p=(3-\sqrt5)/2\), then \(\kappa=2+\sqrt5\), \(\Longrightarrow\) \({\mathcal{M}}\) has type \(\mathrm{III}_{\sqrt5-2}\).
In this section we analyse the centralizer \({\mathcal{M}}^{\omega}:=\{x\in {\mathcal{M}}:\sigma_t^{\omega}(x)=x \text{ for all } t\in{\mathbb{R}}\}\) of state \(\omega\) and the flow of weights for factor \({\mathcal{M}}\).
Lemma 9. For every admissible word \(\xi=(\xi_0,\dots,\xi_n)\in W_n\), \[\mu([\xi])=\pi_0\,p^{\,n+\delta_{\xi_0,1}+\delta_{\xi_n,1}}\kappa^{\,m(\xi)+\delta_{\xi_0,1}},\] where \[m(\xi):=\#\{0\le k<n:\;\xi_k=0,\;\xi_{k+1}=1\}.\] Consequently, if \(\xi,\eta\in W_n\) have the same terminal symbol, then \[\frac{\mu([\xi])}{\mu([\eta])} = p^{\,\delta_{\xi_0,1}-\delta_{\eta_0,1}} \kappa^{\,m(\xi)-m(\eta)+\delta_{\xi_0,1}-\delta_{\eta_0,1}}.\]
Proof. Let \[N_{01}(\xi):=m(\xi), \qquad N_{10}(\xi):=\#\{0\le k<n:\;\xi_k=1,\;\xi_{k+1}=0\},\] and \[N_{00}(\xi):=\#\{0\le k<n:\;\xi_k=0,\;\xi_{k+1}=0\}.\] Since the only admissible transitions are \(00\), \(01\), and \(10\), one has \[N_{01}(\xi)-N_{10}(\xi)=\delta_{\xi_n,1}-\delta_{\xi_0,1},\quadhence\quad N_{10}(\xi)=m(\xi)+\delta_{\xi_0,1}-\delta_{\xi_n,1}.\] Therefore, \[N_{00}(\xi) = n-N_{01}(\xi)-N_{10}(\xi) = n-2m(\xi)-\delta_{\xi_0,1}+\delta_{\xi_n,1}.\] Using \(\pi_1=\pi_0 q\) and \(q=\kappa p^2\), we obtain that \(\mu([\xi])\) equals \[\begin{align} \pi_{\xi_0}\,p^{N_{00}(\xi)}q^{N_{01}(\xi)} =\pi_0\,q^{\delta_{\xi_0,1}} p^{\,n-2m(\xi)-\delta_{\xi_0,1}+\delta_{\xi_n,1}}q^{\,m(\xi)} =\pi_0\,p^{\,n+\delta_{\xi_0,1}+\delta_{\xi_n,1}} \kappa^{\,m(\xi)+\delta_{\xi_0,1}}. \end{align}\] The ratio formula follows immediately. ◻
Proposition 10. For every \(n\geq 0\), set \[{\mathcal{C}}_n := \operatorname{span} \Bigl\{ e^{(n)}_{\xi,\eta}: \xi,\eta\in W_n,\;\xi_n=\eta_n,\;\mu([\xi])=\mu([\eta]) \Bigr\}.\] Then each \({\mathcal{C}}_n\) is a finite-dimensional \(C^*\)-subalgebra of \({\mathcal{A}}_n\), and \[{\mathcal{C}}_n={\mathcal{A}}_n\cap {\mathcal{M}}^{\omega}, \qquad \iota_n({\mathcal{C}}_n)\subset {\mathcal{C}}_{n+1}, \qquad{ {\mathcal{M}}^{\omega}=\left(\operatornamewithlimits{\cup}\limits_{n\geq 0} {\mathcal{C}}_n\right)''}.\] In particular, \({\mathcal{M}}^{\omega}\) is a finite hyperfinite von Neumann algebra, and the restriction \(\omega|_{{\mathcal{M}}^{\omega}}\) is an FNS.
Proof. By Proposition 3, \[\sigma_t^{\omega}\bigl(e^{(n)}_{\xi,\eta}\bigr) = \left(\frac{\mu([\xi])}{\mu([\eta])}\right)^{it} e^{(n)}_{\xi,\eta}, \qquad t\in{\mathbb{R}}.\] Hence \(e^{(n)}_{\xi,\eta}\in{\mathcal{M}}^{\omega}\) if and only if \(\mu([\xi])=\mu([\eta])\). Since the matrix units span \({\mathcal{A}}_n\), it follows that \({\mathcal{C}}_n={\mathcal{A}}_n\cap {\mathcal{M}}^{\omega}\). In particular, each \({\mathcal{C}}_n\) is a finite-dimensional \(C^*\)-subalgebra of \({\mathcal{A}}_n\).
Next, if \(\mu([\xi])=\mu([\eta])\) and \(k\) is an admissible one-step extension of the common terminal symbol \(\xi_n=\eta_n\), then \(\mu([\xi k])=\mu([\xi])P_{\xi_n k} =\mu([\eta])P_{\eta_n k}=\mu([\eta k]).\) Therefore \(\iota_n({\mathcal{C}}_n)\subset {\mathcal{C}}_{n+1}\).
Let \(F_n:{\mathcal{M}}\to{\mathcal{A}}_n\) be the \(\omega\)-preserving normal conditional expectation from Proposition 4. Since \(\sigma_t^{\omega}({\mathcal{A}}_n)={\mathcal{A}}_n\) by Proposition 3, Takesaki’s theorem implies that \(F_n\) commutes with the modular group: \(F_n\circ \sigma_t^{\omega}=\sigma_t^{\omega}\circ F_n\), \(t\in{\mathbb{R}}\). Hence, \(F_n({\mathcal{M}}^{\omega})\subset {\mathcal{A}}_n\cap {\mathcal{M}}^{\omega}={\mathcal{C}}_n\).
Let \((H,\pi,\Omega)\) be the GNS triple of \(({\mathcal{A}},\omega)\), and identify \({\mathcal{M}}\subset{\mathcal{B}}(H)\). For \(x\in{\mathcal{M}}\), define \[P_n(x\Omega):=F_n(x)\Omega.\] As in the proof of Proposition 4, \(P_n\) is the orthogonal projection of \(H\) onto \(\overline{{\mathcal{A}}_n\Omega}\), and therefore \(F_n(y)\Omega\longrightarrow y\Omega\) as \(n\to\infty\) for every \(y\in{\mathcal{M}}\).
Now put \({\mathcal{N}}:=\left(\operatornamewithlimits{\cup}\limits_{n\geq 0} {\mathcal{C}}_n\right)''\). If \(x\in {\mathcal{M}}^{\omega}\), then \(F_n(x)\in {\mathcal{C}}_n\subset{\mathcal{N}}\) for every \(n\). Suppose \(a\in\operatornamewithlimits{\cup}\limits_{m\geq 0}{\mathcal{A}}_m\) and choose \(m_0\) such that \(a\in {\mathcal{A}}_{m_0}\). Then \(a\in{\mathcal{A}}_n\) for every \(n\geq m_0\), and, as \(F_n\) is \({\mathcal{A}}_n\)-bimodular, \[(F_n(x)-x)a\Omega = (F_n(xa)-xa)\Omega \longrightarrow 0.\] Applying the same argument to \(x^*\in {\mathcal{M}}^{\omega}\), we also get \[(F_n(x)-x)^*a\Omega \longrightarrow 0.\] Since \(\operatornamewithlimits{\cup}\limits_{m\geq 0}{\mathcal{A}}_m\Omega\) is dense in \(H\), it follows that \(F_n(x)\) converges strongly* to \(x\). As \({\mathcal{N}}\) is strongly* closed, and \(F_n(x)\in {\mathcal{N}}\), we conclude that \(x\in{\mathcal{N}}\). Thus, \({\mathcal{M}}^{\omega}\subset {\mathcal{N}}\). The reverse inclusion is immediate, because each \({\mathcal{C}}_n\subset {\mathcal{M}}^{\omega}\) and \({\mathcal{M}}^{\omega}\) is weakly closed. Therefore, \({\mathcal{M}}^{\omega}=\left(\operatornamewithlimits{\cup}\limits_{n\geq 0} {\mathcal{C}}_n\right)''\).
Since \({\mathcal{M}}^{\omega}\) is generated by an increasing sequence of finite-dimensional algebras, it is hyperfinite. Finally, if \(x,y\in {\mathcal{M}}^{\omega}\), then \(\sigma_t^{\omega}(x)=x\) for all \(t\in{\mathbb{R}}\), so \(x\) is entire analytic and \(\sigma_{-i}^{\omega}(x)=x\). By the KMS condition, \(\omega(xy)=\omega\bigl(y\,\sigma_{-i}^{\omega}(x)\bigr)=\omega(yx)\). Hence \(\omega|_{{\mathcal{M}}^{\omega}}\) is tracial. It is obviously an FNS, so \({\mathcal{M}}^{\omega}\) is finite. ◻
Remark 11. Under the Feldman–Moore identification of \(({\mathcal{M}},\omega)\) with the measured tail equivalence relation \(({\mathscr R},\mu)\), the algebra \({\mathcal{M}}^{\omega}\) is the Feldman–Moore algebra of the kernel subrelation \[{\mathscr R}_{\mathrm{mod}} = \{(x,y)\in {\mathscr R}:\;D(x,y)=1\}.\] Indeed, the basic compact bisection corresponding to \(e^{(n)}_{\xi,\eta}\) is fixed by the modular group iff \(\mu([\xi])=\mu([\eta])\) \(\Longleftrightarrow\) iff \(D=1\) on that bisection.
Remark 12. Lemma 9 yields a concrete block description of the finite-dimensional algebras \({\mathcal{C}}_n\).
If \(\kappa=1\), then for \(\xi,\eta\in W_n\) with \(\xi_n=\eta_n\), \(\mu([\xi])=\mu([\eta])\), iff \(\xi_0=\eta_0\). Therefore, if \[W_n(i,j):=\{\xi\in W_n:\;\xi_0=i,\;\xi_n=j\}, \qquad i,j\in\{0,1\},\] then \[{\mathcal{C}}_n=\operatornamewithlimits{\oplus}\limits_{i,j\in\{0,1\}}{\mathcal{B}}\bigl(\ell^2(W_n(i,j))\bigr),\] where empty summands are omitted.
Assume that \(\kappa\ne 1\) and \(p\notin \kappa^{{\mathbb{Z}}}\). Then for \(\xi,\eta\in W_n\) with \(\xi_n=\eta_n\), \[\mu([\xi])=\mu([\eta]) \iff \bigl(\xi_0=\eta_0 \text{ and } m(\xi)=m(\eta)\bigr).\] Hence, with \[W_n(i,j,m):=\{\xi\in W_n:\;\xi_0=i,\;\xi_n=j,\;m(\xi)=m\},\] one has \[{\mathcal{C}}_n={\operatornamewithlimits{\oplus}\limits_{i,j\in\{0,1\}}\operatornamewithlimits{\oplus}\limits_{m\geq 0} {\mathcal{B}}\bigl(\ell^2(W_n(i,j,m))\bigr)},\] again omitting empty summands.
Assume that \(\kappa\ne 1\) and \(p=\kappa^\ell\) for some \(\ell\in{\mathbb{Z}}\). Then for \(\xi,\eta\in W_n\) with \(\xi_n=\eta_n\), \[\mu([\xi])=\mu([\eta]) \iff m(\xi)+(\ell+1)\delta_{\xi_0,1} = m(\eta)+(\ell+1)\delta_{\eta_0,1}.\] Thus, if \[E_n(j,r):=\{\xi\in W_n(j):\;m(\xi)+(\ell+1)\delta_{\xi_0,1}=r\}, \qquad j\in\{0,1\},\;r\in{\mathbb{Z}},\] then \[{\mathcal{C}}_n=\operatornamewithlimits{\oplus}\limits_{j=0,1}\operatornamewithlimits{\oplus}\limits_{r\in{\mathbb{Z}}}{\mathcal{B}} \bigl(\ell^2(E_n(j,r))\bigr),\] with empty summands omitted.
Theorem 13. The following assertions hold true:
if \(p\notin \kappa^{\mathbb{Z}}\), then \({\mathcal{M}}^{\omega}\) is not a factor;
if \(p=\kappa^\ell\) for some \(\ell\in\mathbb{Z}\), then \({\mathcal{M}}^{\omega}\) is an HFF of type II\(_1\).
Proof. Set \[{\mathscr R}_{\mathrm{mod}} := \{(x,y)\in {\mathscr R}:\;D(x,y)=1\}.\] By Proposition 10, the centralizer \({\mathcal{M}}^{\omega}\) is finite and hyperfinite. Moreover, as observed above, \({\mathcal{M}}^{\omega}\) is the Feldman–Moore algebra of the measured equivalence relation \({\mathscr R}_{\mathrm{mod}}\). Therefore, \({\mathcal{M}}^{\omega}\) is a factor, i.e., \({\mathscr R}_{\mathrm{mod}}\) is ergodic.
Let \[{\mathscr X}_0:=[0],\qquad {\mathscr X}_1:=[1].\]
(i) Assume that \(p\notin \kappa^{\mathbb{Z}}\). Let \((x,y)\in {\mathscr R}_{\mathrm{mod}}\). Choose \(N\geq 0\) such that \(x_k=y_k\) for all \(k\geq N\), and let \[\xi=(x_0,\dots,x_N),\qquad \eta=(y_0,\dots,y_N).\] Then \(\xi\) and \(\eta\) have the same terminal symbol, and since \((x,y)\in {\mathscr R}_{\mathrm{mod}}\), we have \[1=D(x,y)=\frac{\mu([\xi])}{\mu([\eta])}.\] By Lemma 9, \[\frac{\mu([\xi])}{\mu([\eta])} = p^{\,\delta_{\xi_0,1}-\delta_{\eta_0,1}} \kappa^{\,m(\xi)-m(\eta)+\delta_{\xi_0,1}-\delta_{\eta_0,1}}.\] If \(\xi_0\ne \eta_0\), then \(1=p^{\pm1}\kappa^m\) for some \(m\in\mathbb{Z}\), hence \(p\in\kappa^{\mathbb{Z}}\), contrary to the assumption. Therefore, \(\xi_0=\eta_0\), i.e., \(x_0=y_0\). We have shown that every \({\mathscr R}_{\mathrm{mod}}\)-class is contained either in \({\mathscr X}_0\) or in \({\mathscr X}_1\). Since both \({\mathscr X}_0\) and \({\mathscr X}_1\) have positive \(\mu\)-measure, the relation \({\mathscr R}_{\mathrm{mod}}\) is not ergodic. Hence \({\mathcal{M}}^{\omega}\) is not a factor.
(ii) Now, let us assume \(p=\kappa^\ell\) for some \(\ell\in\mathbb{Z}\). First, we establish that \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\) is ergodic.
In fact, every \(x\in {\mathscr X}_0\) admits a unique decomposition into successive return blocks from one occurrence of symbol \(0\) to the next: \[x=(0,\beta_1,\beta_2,\dots),\;\;\beta_n\in\{a,b\},\quadwhere\;\;a:=0,\;b:=10.\] To make the encoding precise, define the successive return times to the symbol \(0\) by \[T_0(x):=0, \qquad T_{n+1}(x):=\inf\{k>T_n(x): x_k=0\}, \qquad x\in {\mathscr X}_0.\] Since the only admissible transitions are \(00\), \(01\), and \(10\), one has \[T_{n+1}(x)-T_n(x)\in\{1,2\} \qquad (n\geq 0).\] We therefore encode each return block by \[\beta_{n+1}(x):=\begin{cases} a, & T_{n+1}(x)-T_n(x)=1,\\[1mm] b, & T_{n+1}(x)-T_n(x)=2,\end{cases}\;\;where\;\;a:=0,\;b:=10.\] Thus every \(x\in {\mathscr X}_0\) is written uniquely as \(x=(0,\widetilde{\beta}_1,\widetilde{\beta}_2,\dots)\), where \(\widetilde{a}=0\) and \(\widetilde{b}=10\). This defines a bijection \[\Phi:{\mathscr X}_0\to \{a,b\}^{\mathbb{N}}, \qquad \Phi(x)=(\beta_1(x),\beta_2(x),\dots).\]
To see that \(\Phi\) is measurable, let \[C(\beta_1,\dots,\beta_m) := \{\gamma\in\{a,b\}^{\mathbb{N}}:\gamma_1=\beta_1,\dots,\gamma_m=\beta_m\}\] be a cylinder in \(\{a,b\}^{\mathbb{N}}\), and let \(w(\beta_1,\dots,\beta_m)\) be the admissible word obtained by concatenating \(0,\widetilde{\beta}_1,\dots, \widetilde{\beta}_m\). Then \(\Phi^{-1}\bigl(C(\beta_1,\dots,\beta_m)\bigr)=[w(\beta_1,\dots,\beta_m)]\), hence both \(\Phi\) and \(\Phi^{-1}\) are measurable.
Now let \[N_a:=\#\{1\le j\le m:\beta_j=a\}, \qquad N_b:=\#\{1\le j\le m:\beta_j=b\}.\] By the cylinder formula for the Markov measure \(\mu\), \(\mu\bigl([w(\beta_1,\dots,\beta_m)]\bigr) =\mu([0])\,p^{N_a}q^{N_b}\). Indeed, each block \(a\) contributes one transition \(0\to 0\), which has weight \(p\), while each block \(b\) contributes the two-step excursion \(0\to 1\to 0\), which has weight \(q\cdot 1=q\). Therefore, \[\begin{align} \mu\Bigl(\Phi^{-1}\bigl(C(\beta_1,\dots,\beta_m)\bigr)\,\Big|\,{\mathscr X}_0\Bigr) &= \frac{\mu([w(\beta_1,\dots,\beta_m)])}{\mu([0])}= p^{N_a}q^{N_b}= \prod_{j=1}^m \rho(\beta_j), \end{align}\] where \(\rho(a)=p\), \(\rho(b)=q\). Since the cylinders generate the product \(\sigma\)-algebra on \(\{a,b\}^{\mathbb{N}}\), it follows that \(\nu:=\Phi_*\bigl(\mu(\,\cdot\,|\,{\mathscr X}_0)\bigr) =\operatornamewithlimits{\otimes}\limits_{n=1}^\infty (p\,\delta_a+q\,\delta_b)\). In other words, under encoding \(\Phi\), the conditional measure \(\mu(\,\cdot\,|\,{\mathscr X}_0)\) becomes the \((p,q)\) Bernoulli product-measure.
Let \(u\) be any admissible word ending at \(0\). Define \[S_u:[u010]\to [u100], \qquad S_u(u010z)=u100z.\] Since \(\mu([u010])=\mu([u])\,p\,q\) and \(\mu([u100])=\mu([u])\,q\,p\), the Radon–Nikodým derivative of \(S_u\) is equal to \(1\). Hence the graph of \(S_u\) is contained in \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\). In terms of encoding \(\Phi\), this is exactly the adjacent transposition of the two blocks \(a\) and \(b\). Therefore \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\) contains all finite permutations of the block coordinates.
Now let \({\mathscr A}\subset {\mathscr X}_0\) be \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\)-invariant. Then \(\Phi({\mathscr A})\subset \{a,b\}^{\mathbb{N}}\) is invariant under all finite permutations of coordinates. By the Hewitt–Savage zero–one law [30], \(\nu(\Phi({\mathscr A}))\in\{0,1\}\) \(\Longleftrightarrow\) \(\mu({\mathscr A}\,|\,{\mathscr X}_0)\in\{0,1\}\). Hence \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\) is ergodic.
Further, consider \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_1}\). Every \(x\in {\mathscr X}_1\) has the form \(x=(1,0,x_2,x_3,\dots)\). Define \[\theta:{\mathscr X}_1\to {\mathscr X}_0, \qquad \theta(1,0,x_2,x_3,\dots)=(0,x_2,x_3,\dots).\] This is a measure-space isomorphism from \(({\mathscr X}_1,\mu(\,\cdot\,|{\mathscr X}_1))\) onto \(({\mathscr X}_0,\mu(\,\cdot\,|{\mathscr X}_0))\). Moreover, as the initial block \(10\) is common on both sides, \((x,y)\in {\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_1}\) \(\Longleftrightarrow\) \((\theta(x),\theta(y))\in {\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\). Thus \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_1}\) is isomorphic to \({\mathscr R}_{\mathrm{mod}}|_{{\mathscr X}_0}\), and therefore it is ergodic.
We distinguish two cases.
Case 1: \(\ell\geq 0\). Set \[u:=(01)^{\ell+1}0, \qquad v:=10\,0^{\,2\ell+1}.\] Then \(u\) and \(v\) are admissible words of the same length, both end at \(0\), \(u\) starts at \(0\), and \(v\) starts at \(1\). Moreover, \(m(u)=\ell+1\) and \(m(v)=0\). If \(n\) denotes their common length minus \(1\), then Lemma 9 gives \(\mu([u])=\pi_0\,p^n\,\kappa^{\ell+1}\) and \(\mu([v])=\pi_0\,p^{n+1}\kappa\). Since \(p=\kappa^\ell\), these two quantities are equal: \(\mu([u])=\mu([v])\).
Case 2: \(\ell<0\). Write \(\ell=-r\) with \(r\geq 1\), and set \[u:=0^{\,2r}, \qquad v:=1(01)^{r-1}0.\] Again \(u\) and \(v\) are admissible words of the same length, both end at \(0\), \(u\) starts at \(0\), and \(v\) starts at \(1\). Moreover, \(m(u)=0\), and \(m(v)=r-1\). If \(n\) denotes their common length minus \(1\), then Lemma 9 gives \(\mu([u])=\pi_0\,p^n\) and \(\mu([v])=\pi_0\,p^{n+1}\kappa^{r}\). Since \(p=\kappa^{-r}\), we again obtain \(\mu([u])=\mu([v])\).
In both cases, define \(T:[v]\to [u]\), \(T(vz)=uz\). The graph of \(T\) is contained in \({\mathscr R}\), and because \(\mu([u])=\mu([v])\), its Radon–Nikodým derivative is equal to \(1\). Hence the graph of \(T\) is contained in \({\mathscr R}_{\mathrm{mod}}\). Thus \({\mathscr R}_{\mathrm{mod}}\) connects a positive-measure subset of \({\mathscr X}_1\) to a positive-measure subset of \({\mathscr X}_0\).
Now, let us show that \({\mathscr R}_{\mathrm{mod}}\) is ergodic. Let \({\mathscr A}\subset {\mathscr X}\) be \({\mathscr R}_{\mathrm{mod}}\)-invariant. By the above steps, each of \({\mathscr A}\cap {\mathscr X}_0\), \({\mathscr A}\cap {\mathscr X}_1\) has either the zero or the full relative measure in \({\mathscr X}_0\) and \({\mathscr X}_1\), respectively.
Suppose, for contradiction, that \(\mu({\mathscr A}\cap {\mathscr X}_0\,|\,{\mathscr X}_0)=1\) and \(\mu({\mathscr A}\cap {\mathscr X}_1\,|\,{\mathscr X}_1)=0\). Then \({\mathscr A}\cap [u]\) has the full measure in \([u]\). Since the graph of \(T\) is contained in \({\mathscr R}_{\mathrm{mod}}\) and \({\mathscr A}\) is \({\mathscr R}_{\mathrm{mod}}\)-invariant, we have that \(T^{-1}({\mathscr A}\cap [u])\subset {\mathscr A}\cap [v]\). But \(T\) has the Radon–Nikodým derivative \(1\), so \(T^{-1}({\mathscr A}\cap [u])\) has the full measure in \([v]\). Hence \({\mathscr A}\cap [v]\) has a positive measure, contradicting \(\mu({\mathscr A}\cap {\mathscr X}_1\,|\,{\mathscr X}_1)=0\). The opposite mixed case is excluded in the same way. Therefore, \({\mathscr A}\) is either null or conull, so \({\mathscr R}_{\mathrm{mod}}\) is ergodic.
We have proved that \({\mathscr R}_{\mathrm{mod}}\) is ergodic iff \(p\in\kappa^{\mathbb{Z}}\), so \({\mathcal{M}}^{\omega}\) is a factor exactly in that case.
Finally, under the assumption \(p=\kappa^\ell\), Proposition 10 shows that \({\mathcal{M}}^{\omega}\) is finite and hyperfinite. It is also infinite-dimensional, because it contains all diagonal cylinder projections \(e^{(n)}_{\xi,\xi}\). Therefore \({\mathcal{M}}^{\omega}\) is an HFF of type II\(_1\). ◻
Corollary 14. Let \(F({\mathcal{M}})\) be the Connes–Takesaki flow of weights of \({\mathcal{M}}\).
If \(\kappa=1\), then \({\mathcal{M}}\) is of type \(\mathrm{II}_1\), and the flow of weights is trivial.
If \(\kappa\ne 1\), set \(T:=|\log\lambda|=|\log\kappa|\). Then \(F({\mathcal{M}})\) is the periodic translation flow \[{\mathbb{R}}\curvearrowright {\mathbb{R}}/T{\mathbb{Z}}, \quad s\cdot(t+T{\mathbb{Z}})=t+s+T{\mathbb{Z}}.\] Equivalently, \(Z\!\left({\mathcal{M}}\rtimes_{\sigma^{\omega}}{\mathbb{R}}\right)\cong L^\infty({\mathbb{R}}/T{\mathbb{Z}})\).
Proof. If \(\kappa=1\), then Theorem 8 shows that \({\mathcal{M}}\) is of type \(\mathrm{II}_1\), so the flow of weights is trivial. Further, by Theorem 8, for \(\kappa\ne 1\) factor \({\mathcal{M}}\) is of type \(\mathrm{III}_{\lambda}\) with \(0<\lambda<1\). For such a factor, the flow of weights is the periodic translation flow on \({\mathbb{R}}/(|\log\lambda|{\mathbb{Z}})\); see, e.g., [25]. Since \(|\log\lambda|=|\log\kappa|\), the claim follows. ◻
FM thanks Y. Suhov for kind hospitality at DPMMS, University of Cambridge. YS thanks IHES, Bures-sur-Yvette, for hospitality and support.
The author declare that they have no conflict of interests.
Not applicable.