Elias G. Katsoulis, David W. Kribs
We show that the $\ca$-envelope of the tensor algebra of an arbitrary $\ca$-correspondence $\X$ coincides with the Cuntz-Pimsner algebra $Ø_{\X}$, as defined by Katsura \cite{Ka}. This improves earlier results of Muhly and Solel and Fowler, Muhly and Raeburn, who came to the same conclusion under the additional hypothesis that $\X$ is strict and faithful.
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Given an arbitrary countable directed graph $G$ we prove the C*-envelope of the tensor algebra $T_+(G)$ coincides with the universal Cuntz-Krieger algebra associated with $G$. Our approach is concrete in nature and does not rely on Hilbert module machinery. We show how our results extend to the case of higher rank graphs, where an analogous result is obtained for the tensor algebra of a row-finite $k$-graph with no sources.
In this paper we study the C*-envelope of the (non-self-adjoint) tensor algebra associated via subproduct systems to a finite irreducible stochastic matrix $P$. Firstly, we identify the boundary representations of the tensor algebra inside the Toeplitz algebra, also known as its non-commutative Choquet boundary. As an application, we provide examples of C*-envelopes that are not *-isomorphic to either the Toeplitz algebra or the Cuntz-Pimsner algebra. This characterization required a new proof for the fact that the Cuntz-Pimsner algebra associated to $P$ is isomorphic to $C(\mathbb{T}, M_d(\ma…
In this article, we show that the $C^*$-envelope of the minimal tensor product of two operator systems is isomorphic to the minimal tensor product of their $C^*$-envelopes. We do this by identifying the Silov boundary ideal of the minimal tensor product of two operator systems. Finally, as an application of this result, we show that the propagation number of the minimal tensor product of operator systems is the maximum of the propagation numbers of the factors.
We give a new very concrete description of the C*-envelope of the tensor algebra associated to multivariable dynamical system. In the surjective case, this C*-envelope is described as a crossed product by an endomorphism, and as a groupoid C*-algebra. In the non-surjective case, it is a full corner of a such an algebra. We also show that when the space is compact, then the C*-envelope is simple if and only if the system is minimal.
Given a C$^*$-correspondence $X$, we give necessary and sufficient conditions for the tensor algebra $\mathcal T_X^+$ to be hyperrigid. In the case where $X$ is coming from a topological graph we obtain a complete characterization.
Let $\A$ be a unital operator algebra and let $α$ be an automorphism of $\A$ that extends to a *-automorphism of its $\ca$-envelope $\cenv (\A)$. In this paper we introduce the isometric semicrossed product $\A \times_α^{\is} \bbZ^+ $ and we show that $\cenv(\A \times_α^{\is} \bbZ^+) \simeq \cenv (\A) \times_α \bbZ$. In contrast, the $\ca$-envelope of the familiar contractive semicrossed product $\A \times_α \bbZ^+ $ may not equal $\cenv (\A) \times_α \bbZ$. Our main tool for calculating $\ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator al…
Let $(G, P)$ be an abelian, lattice ordered group and let $X$ be a compactly aligned product system over $P$. We show that the C*-envelope of the Nica tensor algebra $\mathcal{N}\mathcal{T}^+_X$ coincides with both Sehnem's covariance algebra $\mathcal{A} \times_X P$ and the co-universal C*-algebra $\mathcal{N}\mathcal{O}^r_X$ for injective, gauge compatible, Nica-covariant representations of Carlsen, Larsen, Sims and Vittadello. We give several applications of this result on both the selfadjoint and non-selfadjoint operator algebra theory. First we guarantee the existence of $\mathcal{N}\math…
Given two correspondences X and Y, we show that (under mild hypotheses) the Cuntz-Pimsner algebra of the tensor product of X and Y embeds as a certain subalgebra of the tensor product of the Cuntz-Pimsner algebra of X and the Cuntz=Pimsner algebra of Y. Furthermore, this subalgebra can be described in a natural way in terms of the gauge actions on the Cuntz-Pimsner algebras. We explore implications for graph algebras, crossed products by the integers, and crossed products by completely positive maps. We also give a new proof of a result of Kaliszewski and Quigg related to coactions on correspo…
We revisit a central result of Muhly and Solel on operator algebras of C*-correspondences. We prove that (possibly non-injective) strongly Morita equivalent C*-correspondences have strongly Morita equivalent relative Cuntz-Pimsner C*-algebras. The same holds for strong Morita equivalence (in the sense of Blecher, Muhly and Paulsen) and strong $Δ$-equivalence (in the sense of Eleftherakis) for the related tensor algebras. In particular, we obtain stable isomorphism of the operator algebras when the equivalence is given by a $σ$-TRO. As an application we show that strong Morita equivalence coinc…
Let $P$ be a submonoid of a group $G$ and let $\mathcal{E}=(\mathcal{E}_p)_{p\in P}$ be a product system over $P$ with coefficient C*-algebra $A$. We show that the following C*-algebras are canonically isomorphic: the C*-envelope of the tensor algebra $\mathcal{T}_λ(\mathcal{E})^+$ of $\mathcal{E}$; the reduced cross sectional C*-algebra of the Fell bundle associated to the canonical coaction of $G$ on the covariance algebra $A\times_{\mathcal{E}}P$ of $\mathcal{E}$; and the C*-envelope of the cosystem obtained by restricting the canonical gauge coaction on $\mathcal{T}_λ(\mathcal{E})$ to the …
Motivated by the theory of tensor algebras and multivariable C*-dynamics, we revisit two fundamental techniques in the theory of C*-correspondences, the "addition of a tail" to a non-injective C*-correspondence and the dilation of an injective C*-correspondence to an essential Hilbert bimodule. We provide a very broad scheme for "adding a tail" to a non-injective C*-correspondence; our scheme includes the "tail" of Muhly and Tomforde as a special case. We illustrate the diversity and necessity of our tails with several examples from the theory of multivariable C*-dynamics. We also exhibit a tr…
We continue the study of isomorphisms of tensor algebras associated to a C*-correspondences in the sense of Muhly and Solel. Inspired by by recent work of Davidson, Ramsey and Shalit, we solve isomorphism problems for tensor algebras arising from weighted partial dynamical systems. We provide complete bounded / isometric classification results for tensor algebras arising from weighted partial systems, both in terms of the C*-correspondences associated to them, and in terms of the original dynamics. We use this to show that the isometric isomorphism and algebraic / bounded isomorphism problems …
$L^p$-modules and $L^p$-correspondences
本文提出 $L^p$-modules 和 $L^p$-correspondences 的理论，涵盖模块、映射及张量积等基本构造。
A cosystem consists of a possibly nonselfadoint operator algebra equipped with a coaction by a discrete group. We introduce the concept of C*-envelope for a cosystem; roughly speaking, this is the smallest C*-algebraic cosystem that contains an equivariant completely isometric copy of the original one. We show that the C*-envelope for a cosystem always exists and we explain how it relates to the usual C*-envelope. We then show that for compactly aligned product systems over group-embeddable right LCM semigroups, the C*-envelope is co-universal, in the sense of Carlsen, Larsen, Sims and Vittade…
Actions of Fell bundles, II: tensor products and Fourier-Stieltjes algebras
本文研究离散群上Fell丛在Hilbert丛上作用的张量积概念，探讨其与C*-对应、类Rieffel诱导过程的联系，并给出Fell丛的Fourier-Stieltjes代数相关结果。
Given two correspondences $X$ and $Y$ and a discrete group $G$ which acts on $X$ and coacts on $Y$, one can define a twisted tensor product $X\boxtimes Y$ which simultaneously generalizes ordinary tensor products and crossed products by group actions and coactions. We show that, under suitable conditions, the Cuntz-Pimsner algebra of this product, $\mathcal O_{X\boxtimes Y}$, is isomorphic to a "balanced" twisted tensor product $\mathcal O_X\boxtimes_\mathbb T\mathcal O_Y$ of the Cuntz-Pimsner algebras of the original correspondences. We interpret this result in several contexts and connect it…
We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando's theorem. Th…
Let $\sS$ be a countable, abelian semigroup of continuous surjections on a compact metric space $X$. Corresponding to this dynamical system we associate two operator algebras, the tensor algebra, and the semicrossed product. There is a unique smallest C$^*$-algebra into which an operator algebra is completely isometrically embedded, which is the C$^*$-envelope. We provide two distinct characterizations of the C$^*$-envelope of the tensor algebra; one developed in a general setting by Katsura, and the other using tools of projective and inductive limits, which gives the C$^*$-envelope as a cros…
We prove that the tensor algebra of a C*-correspondence $X$ is Dirichlet if and only if $X$ is a Hilbert bimodule. As a consequence, we point out and fix an error appearing in the proof of a famous result of Duncan. Secondly we answer a question raised by Davidson and Katsoulis concerning tensor algebras and semi-Dirichlet algebras, by giving an example of a Dirichlet algebra that cannot be described as the tensor algebra of any C*-correspondence. Furthermore we show that the adding tail technique, as extended by the author and Katsoulis, applies in a unique way to preserve the class of Hilber…
We show that if $M$ and $N$ are $C^{*}$-algebras and if $E$ (resp. $F$) is a $C^{*}$-correspondence over $M$ (resp. $N$), then a Morita equivalence between $(E,M)$ and $(F,N)$ implements a isometric functor between the categories of Hilbert modules over the tensor algebras of $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$. We show that this functor maps absolutely continuous Hilbert modules to absolutely continuous Hilbert modules and provides a new interpretation of Popescu's reconstruction operator.
Muhly and Solel developed a notion of Morita equivalence for $C^{*}$- correspondences, which they used to show that if two $C^{*}$-correspondences $E$ and $F$ are Morita equivalent then their tensor algebras $\mathcal{T}_{+}(E)$ and $\mathcal{T}_{+}(F)$ are (strongly) Morita equivalent operator algebras. We give the weak$^{*}$ version of this result by considering (weak) Morita equivalence of $W^{*}$-correspondences and employing Blecher and Kashyap's notion of Morita equivalence for dual operator algebras. More precisely, we show that weak Morita equivalence of $W^{*}$-correspondences $E$ and…
A triangular limit algebra A is isometrically isomorphic to the tensor algebra of a C*-correspondence if and only if its fundamental relation R(A) is a tree admitting a $Z^+_0$-valued continuous and coherent cocycle. For triangular limit algebras which are isomorphic to tensor algebras, we give a very concrete description for their defining C*-correspondence and we show that it forms a complete invariant for isometric isomorphisms between such algebras. A related class of operator algebras is also classified using a variant of the Aho-Hopcroft-Ullman algorithm from computer aided graph theory.
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