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Register a new userC*-envelopes of tensor algebras arising from stochastic matrices
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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 $C(mathbb{T}, M_d(mathbb{C}))$, filling a gap in a previous paper. We then proceed to classify the C*-envelopes of tensor algebras of stochastic matrices up to *-isomorphism and stable isomorphism, in terms of the underlying matrices. This is accomplished by determining the K-theory of these C*-algebras and by combining this information with results due to Paschke and Salinas in extension theory. This classification is applied to provide a clearer picture of the various C*-envelopes that can land between the Toeplitz and the Cuntz-Pimsner algebras.
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We study subproduct systems in the sense of Shalit and Solel arising from stochastic matrices on countable state spaces, and their associated operator algebras. We focus on the non-self-adjoint tensor algebra, and Viselters generalization of the Cuntz-Pimsner C*-algebra to the context of subproduct systems. Suppose that $X$ and $Y$ are Arveson-Stinespring subproduct systems associated to two stochastic matrices over a countable set $Omega$, and let $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ be their tensor algebras. We show that every algebraic isomorphism from $mathcal{T}_+(X)$ onto $mathcal{T}_+(Y)$ is automatically bounded. Furthermore, $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are isometrically isomorphic if and only if $X$ and $Y$ are unitarily isomorphic up to a *-automorphism of $ell^infty(Omega)$. When $Omega$ is finite, we prove that $mathcal{T}_+(X)$ and $mathcal{T}_+(Y)$ are algebraically isomorphic if and only if there exists a similarity between $X$ and $Y$ up to a *-automorphism of $ell^infty(Omega)$. Moreover, we provide an explicit description of the Cuntz-Pimsner algebra $mathcal{O}(X)$ in the case where $Omega$ is finite and the stochastic matrix is essential.
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 are two distinct problems, that require separate criteria to be solved. Our methods yield alternative proofs to classification results for Peters semi-crossed product due to Davidson and Katsoulis and for multiplicity-free graph tensor algebras due to Katsoulis, Kribs and Solel.
Let $A$ be a unital operator algebra and let $alpha$ 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_{alpha}^{is} bbZ^+ $ and we show that $cenv(A times_{alpha}^{is} bbZ^+) simeq cenv (A) times_{alpha} bbZ$. In contrast, the $ca$-envelope of the familiar contractive semicrossed product $A times_{alpha} bbZ^+ $ may not equal $cenv (A) times_{alpha} bbZ$. Our main tool for calculating $ca$-envelopes for semicrossed products is the concept of a relative semicrossed product of an operator algebra, which we explore in the more general context of injective endomorphisms. As an application, we extend a recent result of Davidson and Katsoulis to tensor algebras of $ca$-correspondences. We show that if $T_{X}^{+}$ is the tensor algebra of a $ca$-correspondence $(X, fA)$ and $alpha$ a completely isometric automorphism of $T_{X}^{+}$ that fixes the diagonal elementwise, then the contractive semicrossed product satisfies $ cenv(T_{X}^{+} times_{alpha} bbZ^+)simeq O_{X} times_{alpha} bbZ$, where $O_{X}$ denotes the Cuntz-Pimsner algebra of $(X, fA)$.
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 Sehnems 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 $mathcal{N}mathcal{O}^r_X$, thus settling a problem of Carlsen, Larsen, Sims and Vittadello which was open even for abelian, lattice ordered groups. As a second application, we resolve a problem posed by Skalski and Zacharias on dilating isometric representations of product systems to unitary representations. As a third application we characterize the C*-envelope of the tensor algebra of a finitely aligned higher-rank graph which also holds for topological higher-rank graphs. As a final application we prove reduced Hao-Ng isomorphisms for generalized gauge actions of discrete groups on C*-algebras of product systems. This generalizes recent results that were obtained by various authors in the case where $(G, P) =(mathbb{Z},mathbb{N})$.
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.
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