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Register a new userApproximately diagonalizing matrices over C(Y)
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Huaxin Lin
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Let $X$ be a compact metric space which is locally absolutely retract and let $phi: C(X)to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${rm dim}Yle 2.$ It is proved that there exists a sequence of $n$ continuous maps $alfa_{i,m}: Yto X$ ($i=1,2,...,n$) and a sequence of sets of mutually orthogonal rank one projections ${p_{1, m}, p_{2,m},...,p_{n,m}}subset C(Y, M_n)$ such that $$ lim_{mtoinfty} sum_{i=1}^n f(alfa_{i,m})p_{i,m}=phi(f) for all fin C(X). $$ This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when ${rm dim}Yge 3.$
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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.
Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $theta : Eto F$ is a linear map preserving orthogonality, i.e., $<theta(x), theta(y) > = 0$ whenever $<x, y > = 0$. We show in this article that if, in addition, $A$ has real rank zero, and $theta$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $uin M(A)$ such that $<theta(x), theta(y) > = u < x, y>$ ($x,yin E$). In the case when $A$ is a standard $C^*$-algebra, or when $A$ is a $W^*$-algebra containing no finite type II direct summand, we also obtain the same conclusion with the assumption of $theta$ being an $A$-module map weakened to being a local map.
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.
Frames on Hilbert C*-modules have been defined for unital C*-algebras by Frank and Larson and operator valued frames on a Hilbert space have been studied in arXiv.0707.3272v1.[math.FA]. Goal of the present paper is to introduce operator valued frames on a Hilbert C*-module for a sigma-unital C*-algebra. Theorem 1.4 reformulates the definition given by Frank and Larson in terms of a series of rank-one operators converging in the strict topology. Theorem 2.2. shows that the frame transform and the frame projection of an operator valued frame are limits in the strict topology of a series of elements in the multiplier algebra and hence belong to it. Theorem 3.3 shows that two operator valued frames are right similar if and only if they share the same frame projection. Theorem 3.4 establishes a one to one correspondence between Murray-von Neumann equivalence classes of projections in the multiplier algebra and right similarity equivalence classes of operator valued frames and provides a parametrization of all Parseval operator-valued frames on a given Hilbert C*-module. Left similarity is then defined and Proposition 3.9 establishes when two left unitarily equivalent frames are also right unitarily equivalent.
We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^*$-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph $C^*$-algebras. These include the Toeplitz algebra, Cuntz algebras, and $q$-deformed spheres.
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