1 Introduction↩︎

Alongside the emergence of category theory as a unifying language for abstracting various constructions across Mathematics, determining the precise relationship between two given functorial operations has become crucial for advancing far-reaching structural theories. This is especially true in operator \(K\)-theory and, in particular, Kasparov’s \(KK\)-theory and Higson’s \(E\)-theory. In the setting of \(KK\)-theory, Schochet’s Künneth formula schochet1982topological? is a prime example of this where the relationship between \(K\)-theory and tensor products is determined, and has led to tremendous impact in the classification program for C*-algebras rosenberg1987kunneth?, baum1994classifying?.

In this paper, we resolve the reduced Hao-Ng isomorphism problem in complete generality, establishing the commutation of the reduced crossed product functor with the Cuntz-Pimsner C*-algebra functor. We briefly describe the problem at hand, where more details can be found in Section 2. Given a C*-correspondence \(X\) over a C*-algebra \({\mathcal{B}}\), the smallest gauge-equivariant C*-algebra \({\mathcal{O}}_X\) generated by \(X\) and \({\mathcal{B}}\) is called the Cuntz-Pimsner algebra of \(X\). The Cuntz-Pimsner construction can be largely understood at the level of the underlying \(\mathrm{C}^*\)-correspondence katsura2007ideal?, katsura2004c?, yet it encompasses a wide-variety of naturally occurring \(\mathrm{C}^*\)-algebras, including all crossed products by \({\mathbb{Z}}\) and topological graph algebras katsura2004c?, muhly2005topological?, pimsner1996class?. Given a locally compact Hausdorff group \(G\), a generalized gauge action on \(X\) is an action \(\alpha:G\curvearrowright{\mathcal{O}}_X\) that preserves the copies of \(X\) and \({\mathcal{B}}\) inside \({\mathcal{O}}_X\), and allows us to form the reduced crossed product \(\mathrm{C}^*\)-correspondence \(X\rtimes_{r,\alpha} G\) in a natural way. The reduced Hao-Ng isomorphism then asks whether the Cuntz-Pimsner algebra \({\mathcal{O}}_{X\rtimes_{r,\alpha}G}\) of \(X\rtimes_{r,\alpha}G\) is isomorphic to the corresponding reduced crossed product \({\mathcal{O}}_X \rtimes_{r,\alpha} G\) of the Cuntz-Pimsner algebra of \(X\).

The Hao-Ng isomorphism problem was first considered around 17 years ago by Hao and Ng hao2008crossed? where they established the validity of the isomorphism described above for actions by amenable locally compact Hausdorff groups. In the decade since then, the Hao-Ng isomorphism problems for both full and reduced crossed products were shown to be intimately tied to various functoriality and Takai-type duality for crossed products abadie2010takai?, kaliszewski2013functoriality?, kaliszewski2015coactions? and, in the work of Bédos, Kaliszewski, Quigg, and Robertson bedos2015new?, the Hao-Ng isomorphism for reduced crossed products was established for actions of discrete exact groups. Further applications can be found in other works, including those of Schafhauser on AF-embeddability schafhauser2015cuntz?, and of Deaconu on group actions on graph \(\mathrm{C}^*\)-algebras deaconu2012group?, deaconu2018group?.

In recent years, significant progress has been made on the Hao-Ng isomorphism problems by establishing a bridgehead between the structure theories of C*-algebras and non-self-adjoint operator algebra theory katsoulis2019crossed?, katsoulis2021non?. In these papers, Katsoulis and Ramsey show that to prove the Hao-Ng isomorphism for the reduced crossed product, it is sufficient to prove that the reduced non-self-adjoint operator algebra crossed product functor commutes with the \(\mathrm{C}^*\)-envelope. This strategy was highly successful, and has led to the resolution of the Hao-Ng isomorphism problem for reduced crossed products by discrete groups katsoulis2017c? and under the assumption that the underlying C*-correspondence is hyperrigid katsoulis2021non?.

This tactic of proving commutation results for the C*-envelope was also witnessed by the first-named author together with Geffen and Eilers in dor2020classification?, where it was used to establish a connection between the seemingly distinct classification theories for C*-algebras and non-self-adjoint operator algebras. These types of links between the self-adjoint and non-self-adjoint theories have led to significant structure results in C*-algebra theory, including gauge-coaction co-universality theorems for a variety of C*-algebras dor2020tensor?, dor2022c?, dor2023normal?, kakariadis2023couniversality?, sehnem2022c?, as well as various generalizations of Hao-Ng isomorphism theorems in the context of product systems dor2020tensor?, dor2022c?, katsoulis2020product?, kakariadis2024fock?, li2022zappa?.

In this paper, we establish the commutation of the reduced crossed product functor with the \(\mathrm{C}^*\)-envelope for general locally compact Hausdorff groups.

Over the last decade, the \(\mathrm{C}^*\)-envelope has proven itself as a robust tool in operator algebra theory. It was first shown to exist in Hamana’s work on injective envelopes for operator systems hamana1979injective?, leading to far-reaching consequences in group theory kalantar2017boundaries?, breuillard2017c?, non-commutative convexity theory davidson2017d?, davidson2022strongly?, davidson2019noncommutative?, kennedy2021noncommutative?, and approximation theory bilich2024arveson?, kennedy2015essential?, clouatre2024rigidity?. Theorem 1 is another example of this; allowing us to complete the proof strategy of Katsoulis and Ramsey katsoulis2019crossed?, katsoulis2021non? and resolve the Hao-Ng isomorphism problem for reduced crossed products.

The proof of Theorem 1 is inspired by katsoulis2021non?, where it was established that the reduced crossed product functor commutes with the \(\mathrm{C}^*\)-envelope for hyperrigid operator algebras (see arveson2011noncommutative?). The problem in extending the proof strategy of katsoulis2021non? beyond the hyperrigid setting is rooted in the fact that the unique extension property from non-commutative Choquet theory arveson2008noncommutative? is not preserved under direct integrals. This was first witnessed in a recent counterexample to Arveson’s hyperrigidity conjecture by Bilich and the first-named author bilich2024arveson?.

To overcome this obstruction, our central idea is prompted by another recent paper by Clouâtre with the second-named author clouatre2024rigidity?. Therein, a so-called tight variation of the unique extension property is considered as a substitute for the usual one. Unfortunately, the unique tight extension property does not appear to fit the purpose of proving Theorem 1. Indeed, this is impeded by the general lack of injectivity of the von-Neumann algebra generated by the range of an arbitrary \(*\)-representation of the \(\mathrm{C}^*\)-envelope. This led us in Proposition 2 to consider an intermediate type of unique extension property, where the range of appropriate extensions is contained in \({\mathbb{B}}({\mathcal{H}})\overline{\otimes} L^{\infty}(G)\). However, for our argument to go through beyond the separable setting, we require an operator-valued version of a Maharam-type lifting theorem. A lifting theorem of Ionescu-Tulcea tulcea1967existence? guarantees that there is an appropriate lift for the Lebesgue measure space of a group \(G\) with respect to left Haar measure on \(G\). By applying Hamana’s work on Fubini tensor products hamana1982tensor?, we are able to extend such lifting theorems to the operator-valued setting, which then allows us to prove our main results.

This paper has four sections, including this introduction. In Section 2, we gather preliminary facts on the \(\mathrm{C}^*\)-envelope, the injective envelope, C*-correspondences, and Cuntz-Pimsner algebras. In Section 3, we prove operator-valued Maharam-type lifting theorems through the use of Hamana’s Fubini tensor products. In Section 4, we establish the commutation of the C*-envelope with the reduced crossed product (Theorem 1), which leads to the resolution of the Hao-Ng isomorphism theorem for reduced crossed products (Theorem 2).

2 Preliminaries↩︎

2.1 The C*-envelope and injective envelope↩︎

We recall some of the necessary machinery from operator algebra theory, which may be found in arveson2008noncommutative?, paulsen2002completely?. An operator algebra is a norm-closed subalgebra of bounded operators on a Hilbert space \({\mathcal{A}}\subseteq {\mathbb{B}}({\mathcal{H}})\). A representation of \({\mathcal{A}}\) is a completely contractive homomorphism \(\rho:{\mathcal{A}}\rightarrow{\mathbb{B}}({\mathcal{K}})\). The \(\mathrm{C}^*\)-envelope of \({\mathcal{A}}\) is a pair \((\mathrm{C}^*_e({\mathcal{A}}), \varepsilon)\) consisting of a \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*_e({\mathcal{A}})\) together with a completely isometric representation \(\varepsilon:{\mathcal{A}}\rightarrow\mathrm{C}^*_e({\mathcal{A}})\) such that \(\mathrm{C}^*(\varepsilon({\mathcal{A}})) = \mathrm{C}^*_e({\mathcal{A}})\) has the following co-universal property: whenever \(\iota:{\mathcal{A}}\rightarrow{\mathcal{B}}\) is a completely isometric representation with \({\mathcal{B}}= \mathrm{C}^*(\iota({\mathcal{A}}))\), then there is a surjective \(*\)-homomorphism \(\pi:{\mathcal{B}}\rightarrow\mathrm{C}^*_e({\mathcal{A}})\) such that \(\pi\circ\iota = \varepsilon\). Since this property uniquely determines the C*-envelope up to a \(*\)-isomorphism that preserves an image of \({\mathcal{A}}\), we frequently refer to the \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*_e({\mathcal{A}})\) as the \(\mathrm{C}^*\)-envelope of \({\mathcal{A}}\).

In Arveson’s original paper on the subject arveson1969subalgebras?, a method of constructing the \(\mathrm{C}^*\)-envelope of a unital operator algebra \({\mathcal{A}}\subseteq {\mathbb{B}}({\mathcal{H}})\) was proposed. This method is different from the one used by Hamana, and is more in-line with classical Choquet theory. For this, suppose that \({\mathcal{B}}:=\mathrm{C}^*({\mathcal{A}})\) is the \(\mathrm{C}^*\)-algebra generated by \({\mathcal{A}}\) in \({\mathbb{B}}({\mathcal{H}})\). A unital \(*\)-representation \(\pi:{\mathcal{B}}\rightarrow{\mathbb{B}}({\mathcal{K}})\) is said to have the unique extension property with respect to \({\mathcal{A}}\) if there is a unique completely positive map \(\psi:{\mathcal{B}}\rightarrow{\mathbb{B}}({\mathcal{K}})\) such that \(\psi|_{\mathcal{A}}= \pi|_{\mathcal{A}}\). This is a non-commutative analogue of having the unique representing measure for a point evaluation be Dirac mass at that point. Now, if \(\pi:{\mathcal{B}}\rightarrow {\mathbb{B}}({\mathcal{K}})\) has the unique extension property with respect to \({\mathcal{A}}\) and \(\pi|_{\mathcal{A}}\) is completely isometric, then it is easy to show that \((\pi({\mathcal{B}}), \pi|_{\mathcal{A}})\) coincides with the \(\mathrm{C}^*\)-envelope of \({\mathcal{A}}\) (see arveson1969subalgebras?). The existence of such \(*\)-representations was first exhibited by Dritschel and McCullough dritschel2005boundary?, and has led to the resolution of Arveson’s conjecture on sufficiency of boundary representations arveson2008noncommutative?, davidson2015choquet?.

Throughout this paper, we shall consider non-unital operator algebras. However, in practice, our operator algebras will possess a self-adjoint contractive approximate unit. The \(\mathrm{C}^*\)-envelope of a non-unital operator algebra is also known to exist and behave similarly to the one in the unital setting. Given a non-unital operator algebra \({\mathcal{A}}\), we let \({\mathcal{A}}^\sim := {\mathcal{A}}\oplus \mathbb{C}I_{{\mathcal{H}}}\) denote the one-point unitization of \({\mathcal{A}}\). By a theorem of Meyer meyer2001adjoining?, \({\mathcal{A}}^\sim\) is uniquely determined up to a unital completely isometric isomorphism of \({\mathcal{A}}\subseteq {\mathbb{B}}({\mathcal{H}})\). Furthermore, if \((\mathrm{C}^*_e({\mathcal{A}}^\sim), \varepsilon)\) is the \(\mathrm{C}^*\)-envelope of \({\mathcal{A}}^\sim\), the \(\mathrm{C}^*\)-envelope of \({\mathcal{A}}\) may be identified with the \(\mathrm{C}^*\)-algebra generated by \(\varepsilon({\mathcal{A}})\) blecher2004operator? inside \(\mathrm{C}^*_e({\mathcal{A}}^\sim)\). For further details on the unique extension property of representations of non-unital operator algebras, we refer the reader to dor2018full?.

We will also require some of the theory on injective envelopes of operator systems. We refer the reader to paulsen2002completely? for further details. An operator system is a unital self-adjoint subspace \({\mathcal{E}}\subseteq {\mathbb{B}}({\mathcal{H}})\), and \({\mathcal{E}}\) is said to be injective if it is an injective object in the category of operator systems with unital completely positive maps as morphisms. An injective envelope of an operator system \({\mathcal{S}}\subseteq {\mathbb{B}}({\mathcal{H}})\) is a pair \((I({\mathcal{S}}), \iota)\) consisting of an injective operator system \(I({\mathcal{S}})\) and a unital completely isometric map \(\iota:{\mathcal{S}}\rightarrow I({\mathcal{S}})\) with the property that whenever \({\mathcal{E}}\) is an injective operator system such that \({\mathcal{S}}\subseteq {\mathcal{E}}\subseteq I({\mathcal{S}})\), then \({\mathcal{E}}= I({\mathcal{S}})\). Any operator system has an essentially unique injective envelope, as was first shown by Hamana in hamana1979injective?. If \({\mathcal{A}}\) is a unital operator algebra, then \({\mathcal{A}}\) also admits an injective envelope and \(I({\mathcal{A}}) = I({\mathcal{A}}+{\mathcal{A}}^*)\) paulsen2002completely?.

Any injective operator system \({\mathcal{E}}\subseteq {\mathbb{B}}({\mathcal{H}})\) gives rise to a unital completely positive idempotent \(\theta:{\mathbb{B}}({\mathcal{H}})\rightarrow {\mathbb{B}}({\mathcal{H}})\) whose range is equal to \({\mathcal{E}}\). Through this UCP idempotent, \({\mathcal{E}}\) may be equipped with a Choi–Effros multiplication \(s\cdot_\theta t := \theta(st)\), which gives \({\mathcal{E}}\) the structure of a \(\mathrm{C}^*\)-algebra paulsen2002completely?. For the injective envelope of a unital opreator algebra, the \(\mathrm{C}^*\)-subalgebra of \((I({\mathcal{A}}), \cdot_\theta)\) generated by \({\mathcal{A}}\) coincides with \(\mathrm{C}^*_e({\mathcal{A}})\) paulsen2002completely?. Thus, when \({\mathcal{A}}\) is non-unital, the \(\mathrm{C}^*\)-subalgebra of \((I({\mathcal{A}}^{\sim}), \cdot_\theta)\) generated by \({\mathcal{A}}\) will also coincide with \(\mathrm{C}^*_e({\mathcal{A}})\).

Let \({\mathcal{S}}\) be an operator system. A UCP idempotent \(\gamma:{\mathbb{B}}({\mathcal{H}})\rightarrow{\mathbb{B}}({\mathcal{H}})\) that fixes \({\mathcal{S}}\) is called an \({\mathcal{S}}\)-projection, and the space of \({\mathcal{S}}\)-projections is partially ordered by \(\prec\) where \(\theta\prec\gamma\) if and only if \(\theta\circ\gamma = \gamma\circ\theta = \theta\). If \(\theta\) is a minimal \({\mathcal{S}}\)-projection, then \((\theta({\mathbb{B}}({\mathcal{H}})), \operatorname{id}_{\mathcal{S}})\) is a copy of the injective envelope of \({\mathcal{S}}\) paulsen2011weak?. In fact, an application of the Ellis lemma (see furstenberg1989idempotents?) shows that if \(\gamma : {\mathbb{B}}({\mathcal{H}}) \rightarrow {\mathbb{B}}({\mathcal{H}})\) is an \({\mathcal{S}}\)-projection, there always exists a minimal \({\mathcal{S}}\)-projection \(\theta:{\mathbb{B}}({\mathcal{H}}) \rightarrow {\mathbb{B}}({\mathcal{H}})\) with \(\theta \prec \gamma\). In other words, we can always guarantee that a copy of \(I({\mathcal{S}})\) is contained in any given injective C*-algebra \(\gamma({\mathbb{B}}({\mathcal{H}}))\).

2.2 C*-correspondences, Cuntz-Pimsner algebras, and their dynamics↩︎

Here, we record some of the details on \(\mathrm{C}^*\)-correspondences that we shall need in this paper, and refer the reader to brown2008textrm? or lance1995hilbert? for more. Let \({\mathcal{B}}\) be a \(\mathrm{C}^*\)-algebra. For a right Hilbert \({\mathcal{B}}\)-module \(X\), we denote by \({\mathcal{L}}(X)\) the \(\mathrm{C}^*\)-algebra of adjointable operators and by \({\mathcal{K}}(X)\) its subalgebra given by the closure of the “rank-one" operators. More precisely, for \(x,y\in X\), the rank-one operators are denoted by \(\theta_{x,y}\) and defined by \(\theta_{x,y}(z) := x\langle y, z\rangle\).

Recall that a \(\mathrm{C}^*\)-correspondence over \({\mathcal{B}}\) is a right Hilbert \({\mathcal{B}}\)-module \(X\) together with a \(*\)-homomorphism \(\varphi_X: {\mathcal{B}}\rightarrow{\mathcal{L}}(X)\), which we consider as a left action of \({\mathcal{B}}\) on \(X\). We will assume throughout this paper that C*-correspondences are non-degenerate, where we say that \(X\) is non-degenerate if the \(\varphi_X({\mathcal{B}})X\) is dense in \(X\).

Given two \(\mathrm{C}^*\)-correspondences \(X\) and \(Y\), a new \(\mathrm{C}^*\)-correspondence \(X\otimes Y\) may be formed in the following way: First, on the quotient of the algebraic tensor product \(X\odot Y\) by \[xb\otimes y - x\otimes \varphi_Y(b)y, \quad x\in X, y\in Y, b\in {\mathcal{B}},\] we may define a \({\mathcal{B}}\)-valued sesquilinear form and a two-sided \({\mathcal{B}}\)-action by setting \[\langle x\otimes y, v\otimes w\rangle = \langle y, \varphi_Y(\langle x, v\rangle)w\rangle,\] \[(x\otimes y)b = x\otimes(yb), \quad \text{and} \quad \varphi_{X\otimes Y}(b)(x\otimes y) = (\varphi_X(b)x)\otimes y\] for every \(v,x\in X, w,y\in Y\) and \(b\in{\mathcal{B}}\). Then, the completion with respect to the \({\mathcal{B}}\)-valued inner product yields a \(\mathrm{C}^*\)-correspondence \(X\otimes Y\).

For a \(\mathrm{C}^*\)-correspondence \(X\), one may then construct the Fock correspondence \[{\mathcal{F}}_X = {\mathcal{B}}\oplus \bigoplus_{n=1}^{\infty} X^{\otimes n}\] over \({\mathcal{B}}\). The Fock correspondence gives rise to the so-called left-creation operators \(T_x\in{\mathcal{L}}({\mathcal{F}}_X)\) for \(x\in X,\) defined by \[T_x(b) = xb \quad \text{and} \quad T_x(x_1\otimes\ldots\otimes x_n) = x\otimes x_1\otimes\ldots\otimes x_n\] for \(b\in{\mathcal{B}}\) and \(x, x_1,\ldots, x_n\in X\). The \(\mathrm{C}^*\)-algebra \({\mathcal{T}}_X\) generated by the left action of \({\mathcal{B}}\) on \({\mathcal{F}}_X\) and the left-creation operators on \({\mathcal{F}}_X\) is called the Toeplitz algebra of the \(\mathrm{C}^*\)-correspondence \(X\). The norm-closed subalgebra \({\mathcal{T}}_X^+\) generated by the left action of \({\mathcal{B}}\) on \({\mathcal{F}}_X\) and the left-creation operators is called the tensor algebra of \(X\).

Let \({\mathcal{B}}, {\mathcal{C}}\) be \(\mathrm{C}^*\)-algebras and \(X\) be a \(\mathrm{C}^*\)-correspondence over \({\mathcal{B}}\). A representation of \(X\) is a pair \((\rho, t)\) consisting of a non-degenerate \(*\)-homomorphism \(\rho:{\mathcal{B}}\rightarrow {\mathcal{C}}\) and a completely contractive linear map \(t:X\rightarrow {\mathcal{C}}\) with the property that \[\rho(a)t(x)\rho(b) = t(\varphi_X(a)x\varphi_X(b)), \quad a,b\in {\mathcal{B}}, x\in X.\] If, in addition, we have \(t(x)^*t(y) = \rho(\langle x, y\rangle)\) for each \(x,y\in X\), then we say that \((\rho, t)\) is rigged (this is sometimes called isometric, for instance in muhly1998tensor?). In pimsner1996class?, muhly1998tensor?, it is shown that the Toeplitz algebra \({\mathcal{T}}_X\) is the universal \(\mathrm{C}^*\)-algebra generated by the rigged representations of \(X\).

Given a C*-correspondence \(X\) over \({\mathcal{B}}\), we define Katsura’s ideal \(J_X \lhd {\mathcal{B}}\) by \[J_X:= \{ \;b\in {\mathcal{B}}\;| \;\varphi_X(b) \in {\mathcal{K}}(X) \;\;\text{and} \;\;bc = 0 \;\;\text{for all } c\in \ker \varphi_X \;\}.\] Now, given a rigged representation \((\rho, t)\) of \(X\), one may define a \(*\)-homomorphism \(\psi_t:{\mathcal{K}}(X)\rightarrow {\mathcal{C}}\) by specifying it on rank-one operators by \(\psi_t(\theta_{x,y}) = t(x)^*t(y)\) for each \(x,y\in X\). A rigged representation \((\rho, t)\) is then said to be covariant if \(\psi_t(\varphi_X(b)) = \rho(b)\) for each \(b\in J_X\). The universal \(\mathrm{C}^*\)-algebra generated by rigged covariant representations of \(X\) is the Cuntz-Pimsner algebra \({\mathcal{O}}_X\), introduced by Pimsner pimsner1996class? and refined by Katsura katsura2004c?. Thus, if \((\hat{\rho},\hat{t})\) is a universal rigged representation (so that the C*-algebra generated by the images of \(\hat{\rho}\) and \(\hat{t}\) is a \(*\)-isomorphic copy of \({\mathcal{T}}_X\)), we see that \({\mathcal{O}}_X\) is the quotient of \({\mathcal{T}}_X\) by the ideal generated by the differences \(\psi_{\hat{t}}(\varphi_X(b)) - \hat{\rho}(b)\) for \(b\in J_X\). In fact, by katsoulis2006tensor?, we know that \({\mathcal{O}}_X\) can be identified with the C*-envelope \(\mathrm{C}^*_e({\mathcal{T}}_X^+)\). In particular, the canonical quotient map from \({\mathcal{T}}_X\) to \({\mathcal{O}}_X\) is completely isometric on \({\mathcal{T}}_X^+\). So, the tensor algebra can be thought of as an operator subalgebra in both \({\mathcal{T}}_X\) and \({\mathcal{O}}_X\).

Given a locally compact Hausdorff group \(G\), a generalized gauge action \(\alpha:G\curvearrowright X\) is an action \(\alpha\) of \(G\) on \({\mathcal{T}}_X^+\) such that \(\alpha_g({\mathcal{B}}) = {\mathcal{B}}\) and \(\alpha_g(X) = X\) for every \(g\in G\). Equivalently, a generalized gauge action is the restriction of an action \(\alpha\) of \(G\) on \({\mathcal{T}}_X\), or \({\mathcal{O}}_X\), to \({\mathcal{T}}_X^+\) such that \(\alpha_g({\mathcal{B}}) = {\mathcal{B}}\) and \(\alpha_g(X) = X\) for each \(g\in G\) with respect to the canonical copies of \({\mathcal{B}}\) and \(X\) in each of these generated C*-algebras. By the above description of the Cuntz-Pimsner algebra, we see that \(\alpha\) automatically induces an action on \({\mathcal{O}}_X\), which we will continue to denote by \(\alpha\). For a generalized gauge action, one may construct a \(\mathrm{C}^*\)-correspondence \(X\rtimes_{\alpha, r} G\) over \({\mathcal{B}}\rtimes_{\alpha, r}G\) (see bedos2015new? for an equivalent definition) by taking the closures of \(\mathrm{C}_c(G, X)\) and \(\mathrm{C}_c(G, {\mathcal{B}})\) considered canonically inside \({\mathcal{T}}_X\rtimes_{\alpha, r}G\) (which is itself a completion of \(\mathrm{C}_c(G,{\mathcal{T}}_X)\)), where the bimodule actions are induced by multiplication and the \(({\mathcal{B}}\rtimes_{\alpha, r}G)\)-valued inner product is given by \(\langle f, g\rangle = f^*g\) for \(f,g\in\mathrm{C}_c(G, X)\).

3 Operator-valued Maharam-type lifting theorems↩︎

For our arguments, we require generalizations of classical measure-theoretic lifting theorems. To this end, let \((X, {\mathcal{F}}, \mu)\) be a complete measure space and \(M^\infty({\mathcal{F}})\) denote the space of bounded measurable functions on \(X\), which is a C*-subalgebra of all bounded functions \(\ell^{\infty}(X)\). Then, there is a surjective \(*\)-homomorphism \(q:M^\infty({\mathcal{F}})\rightarrow L^\infty(X, \mu)\) defined by identifying functions \(\mu\)-a.e. A lifting for \((X, {\mathcal{F}}, \mu)\) is a unital \(*\)-homomorphism \(\rho: L^\infty(X, \mu)\rightarrow M^\infty({\mathcal{F}})\) such that \(q\circ\rho = \operatorname{id}\). The lifting theorem of Maharam maharam1958theorem? states that \((X, {\mathcal{F}}, \mu)\) always admits a lifting when \(\mu\) is a finite measure. The existence of a lift is known in several other circumstances tulcea2012topics?, including the measure space \((G,{\mathcal{F}},\mu)\) where \(G\) is a locally compact Hausdorff group, \(\mu\) is left Haar measure, and \({\mathcal{F}}\) is the completion of Borel \(\sigma\)-algebra on \(G\) with respect to \(\mu\) tulcea1967existence?.

Our lifting result will apply Hamana’s work on Fubini tensor products hamana1982tensor?. For this, fix a pair of Hilbert spaces \({\mathcal{H}}, {\mathcal{K}}\) and let \(\{e_\alpha : \alpha\in A\}\) be an orthonormal basis for \({\mathcal{H}}\). For \(\alpha,\beta\in A\), we define rank-one operators on \({\mathcal{H}}\) by \(E_{\alpha\beta}(h) = \langle h, e_\beta\rangle e_\alpha\). Furthermore, for each \(\alpha\in A\), let \(J_\alpha: {\mathcal{K}}\rightarrow{\mathcal{H}}\otimes{\mathcal{K}}\) be the isometry defined by \(J_\alpha (k) = e_\alpha\otimes k\). Then, each \(x\in {\mathbb{B}}({\mathcal{H}}\otimes{\mathcal{K}})\) can be expressed as \[x = \sum_{\alpha, \beta\in A} E_{\alpha\beta}\otimes J_\alpha^* xJ_\beta\] where the sum converges in the strong operator topology. Let \({\mathcal{S}}\subseteq {\mathbb{B}}({\mathcal{K}})\) be a norm-closed operator system. Following hamana1982tensor?, we may form an operator system \[{\mathbb{B}}({\mathcal{H}})\overline{\otimes}{\mathcal{S}}: = \{x \in {\mathbb{B}}({\mathcal{H}}\otimes{\mathcal{K}}) ~:~ (J_\alpha^* xJ_\beta)\in{\mathcal{S}}\text{ for all \alpha,\beta\in A}\}.\] When \({\mathcal{M}}\) is a von Neumann algebra, \({\mathbb{B}}({\mathcal{H}})\overline{\otimes}{\mathcal{M}}\) agrees with the usual von Neumann algebra tensor product hamana1982tensor?.

From hamana1982tensor?, we can guarantee that there is a unique unital completely positive map \[\operatorname{id}\overline{\otimes}q:{\mathbb{B}}({\mathcal{H}})\overline{\otimes} M^\infty({\mathcal{F}}) \rightarrow {\mathbb{B}}({\mathcal{H}})\overline{\otimes}L^\infty(X, \mu)\] that extends \(\operatorname{id}\odot q\). In fact, we will show that whenever a lifting for \((X, {\mathcal{F}}, \mu)\) exists, the map \(\operatorname{id}\overline{\otimes}q\) has a natural unital completely positive right inverse. Before proceeding with the proof, we record some relevant information. Recall that a \(\mathrm{C}^*\)-algebra \({\mathcal{B}}\) is said to be monotone complete (respectively, \(\sigma\)-monotone complete) if every increasing bounded net (sequence) of self-adjoint operators in \({\mathcal{B}}\) has a least upper bound in \({\mathcal{B}}\).

In hamana1982tensor?, Hamana proved that \({\mathbb{B}}({\mathcal{H}})\overline{\otimes} {\mathcal{B}}\) is a monotone complete \(\mathrm{C}^*\)-algebra whenever \({\mathcal{B}}\) is. Furthermore, in saito2016tensor?, Saitô showed that when \({\mathcal{H}}\) is separable, and \({\mathcal{B}}\) is \(\sigma\)-monotone complete, then \({\mathbb{B}}({\mathcal{H}})\overline{\otimes} {\mathcal{B}}\) is also a \(\sigma\)-monotone complete C*-algebra. For our case of interest, \(M^\infty({\mathcal{F}})\) is merely \(\sigma\)-monotone complete and we will need \({\mathcal{H}}\) to be potentially non-separable. Thus, at least a priori, we only know that \({\mathbb{B}}({\mathcal{H}})\overline{\otimes}M^\infty({\mathcal{F}})\) is an operator system. This is an important distinction, as there are examples of commutative \(\mathrm{C}^*\)-algebras acting on non-separable Hilbert space for which \({\mathbb{B}}({\mathcal{H}})\overline{\otimes}{\mathcal{B}}\) is not a \(\mathrm{C}^*\)-algebra saito2016tensor?.

Although the operator system structure will be sufficient for our purposes, a finer analysis reveals that \(\operatorname{id}\overline{\otimes}\rho\) still remains multiplicative, at least in some sense. Indeed, \({\mathcal{C}}:=\rho(L^\infty(X, \mu))\) is monotone complete as it is an injective \(\mathrm{C}^*\)-algebra hamana1982tensor?. Thus, by hamana1982tensor? we find that \({\mathbb{B}}({\mathcal{H}})\overline{\otimes}{\mathcal{C}}\) is a monotone complete \(\mathrm{C}^*\)-algebra and \(\operatorname{id}\overline{\otimes}\rho : {\mathbb{B}}({\mathcal{H}})\overline{\otimes} L^\infty(X, \mu)\rightarrow {\mathbb{B}}({\mathcal{H}})\overline{\otimes} {\mathcal{C}}\) is a unital injective \(*\)-homomorphism, where \({\mathbb{B}}({\mathcal{H}})\overline{\otimes} {\mathcal{C}}\) is given the Choi-Effros product. Thus, it must automatically be normal by the monotone completeness of its domain and range. Furthermore, the inclusion mapping \({\mathcal{C}}\subseteq M^\infty({\mathcal{F}})\) promotes to a unital completely isometric map \(\operatorname{id}\overline{\otimes}\operatorname{id}: {\mathbb{B}}({\mathcal{H}})\overline{\otimes}{\mathcal{C}}\rightarrow {\mathbb{B}}({\mathcal{H}})\overline{\otimes}M^\infty({\mathcal{F}})\) that uniquely extends \(\operatorname{id}\odot\operatorname{id}\) by hamana1982tensor?. The uniqueness of such a map ensures that this must be the identity map. Thus, in this sense we may regard \(\operatorname{id}\overline{\otimes}\rho: {\mathbb{B}}({\mathcal{H}})\overline{\otimes} L^\infty(X, \mu)\rightarrow {\mathbb{B}}({\mathcal{H}}) \overline{\otimes} M^\infty({\mathcal{F}})\) as a unital normal injective \(*\)-homomorphism, even though \({\mathbb{B}}({\mathcal{H}}) \overline{\otimes} M^\infty({\mathcal{F}})\) is not known to be a C*-algebra in general.

4 Reduced crossed products and their C*-envelopes↩︎

Let \(G\) be a locally compact Hausdorff group and \(m\) denote the left Haar measure on \(G\). Consider an approximately unital operator algebra \({\mathcal{A}}\) and a point-norm continuous action \(\alpha:G\curvearrowright{\mathcal{A}}\) by completely isometric automorphisms. Throughout, we will refer to the triple \(({\mathcal{A}}, G, \alpha)\) as a dynamical system. If \({\mathcal{A}}\) happens to be a C*-algebra, we will refer to the triple \(({\mathcal{A}}, G, \alpha)\) as a C*-dynamical system.

A completely contractive representation \(\pi:{\mathcal{A}}\rightarrow{\mathbb{B}}({\mathcal{H}})\) is said to be \(\alpha\)-admissible if there is an action \(\widehat{\alpha} : G \curvearrowright \mathrm{C}^*(\pi(\mathcal{A}))\) such that \(\widehat{\alpha}_g \circ \pi = \pi \circ \alpha_g\). When the \(\alpha\)-admissible representation \(\pi\) is completely isometric and considered as an embedding, we will abuse notation and continue to denote the extended action \(\widehat{\alpha}\) by \(\alpha\). A dynamical system always admits a completely isometric \(\alpha\)-admissible embedding. For instance, the embedding \(\varepsilon:{\mathcal{A}}\rightarrow\mathrm{C}^*_e({\mathcal{A}})\) is always \(\alpha\)-admissible katsoulis2019crossed?. A covariant pair \((\pi, \mu)\) for \(({\mathcal{A}}, G, \alpha)\) consists of a non-degenerate representation \(\pi:{\mathcal{A}}\rightarrow{\mathbb{B}}({\mathcal{H}})\) and a strongly continuous representation \(\mu: G \rightarrow{\mathbb{B}}({\mathcal{H}})\) such that \[\pi(\alpha_g(a)) = \mu_g \pi(a)\mu_g^*, \quad g\in G.\]

Note that whenever \((\pi,\mu)\) is a covariant pair, it follows that \(\pi\) is automatically \(\alpha\)-admissible where the action \(\widehat{\alpha}\) of \(G\) on \(C^*(\pi(A))\) is given by \(\widehat{\alpha}(T) = \mu_gT\mu_g^*\). Now, given a non-degenerate representation \(\pi : {\mathcal{A}}\rightarrow {\mathbb{B}}({\mathcal{H}})\), we may form the non-degenerate representation \(\pi_{\alpha} : {\mathcal{A}}\rightarrow L^{\infty}(G; {\mathbb{B}}({\mathcal{H}})) \cong {\mathbb{B}}({\mathcal{H}}) \overline{\otimes} L^{\infty}(G) \subseteq {\mathbb{B}}({\mathcal{H}}\otimes L^2(G))\) by setting \(\pi_{\alpha}(a)(g) := (\pi \circ \alpha_g^{-1})(a)\) for \(g\in G\). Moreover, note that since \(g \mapsto \pi \circ \alpha_g\) is point-norm continuous, the image of \(\pi_{\alpha}\) is actually contained in \(C_b(G;{\mathbb{B}}({\mathcal{H}}))\). We let \(\lambda : G \rightarrow {\mathbb{B}}(L^2(G))\) be the left-regular representation of \(G\), so that \(\operatorname{id}\otimes \lambda\) is a strongly continuous representation of \(G\) on \({\mathcal{H}}\otimes L^2(G)\). Then, the pair \((\pi_{\alpha},\operatorname{id}\otimes \lambda)\) is readily verified to be covariant. In particular, \(\pi_{\alpha}\) is automatically \(\alpha\)-admissible.

Given a dynamical system \(({\mathcal{A}},G,\alpha)\) and a covariant pair \((\pi,\mu)\), one may form the integrated form \(\pi \rtimes \mu\) on \(C_c(G,{\mathcal{A}})\) given by \([\pi \rtimes \mu](f) = \int_G \pi(f(g))\mu_g dm(g)\), where the latter integral is understood as in williams2007book?. Since the reduced crossed product for \(({\mathcal{A}}, G, \alpha)\) is independent of the choice of a completely isometric \(\alpha\)-admissible \(\pi\) katsoulis2019crossed?, we may then define the reduced crossed product as follows.

The following proposition is the key observation that allows us to push the strategy of katsoulis2021non?, and verify the commutation of the \(\mathrm{C}^*\)-envelope with the reduced crossed product. In what follows, we recall dor2018full? where the unique extension property for potentially non-unital operator algebras is defined with respect to completely contractive completely positive extensions to a generated C*-algebra.

With this at hand, we may now refine katsoulis2021non?. This provides a complete answer to a problem that was first considered in katsoulis2019crossed? when the operator algebra possess a self-adjoint contractive approximate identity.

Now that we are equipped with Theorem 3, we are at the stage where we can settle the Hao-Ng isomorphism problem for reduced crossed products.

Although the strategy presented in katsoulis2021non? is shown here to be successful in resolving the reduced Hao-Ng isomorphsm in complete generality, the corresponding full Hao-Ng isomorphism is more difficult. Indeed, it is known that the full Hao-Ng isomorphism is equivalent to the commutation of the \(\mathrm{C}^*\)-envelope and the full crossed product for tensor algebras katsoulis2021non?. As before, it would suffice to prove an analogue of Theorem 3 for full crossed products. Unfortunately, this fails for arbitrary operator algebras harris2019crossed?, while it is unknown whether it is valid for the class of tensor algebras. Thus, a more nuanced approach would be necessary to resolve the full Hao-Ng isomorphism in complete generality.

Acknowledgments. The authors are grateful to Boyu Li for several fruitful discussions held at New Mexico State University and electronically over Zoom, and to Boris Bilich and Jamie Gabe for suggestions and comments on the paper. The authors are especially grateful to Raphaël Clouâtre for pointing out a missing argument in a preliminary version of the paper.