Algebraic isomorphisms of limit algebras

Donsig, Allan P.; Hudson, Timothy D.; Katsoulis, Elias G.

doi:10.1090/s0002-9947-00-02714-8

Cited by 11 publications

(11 citation statements)

References 22 publications

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“…It is readily verified that the graph of ϕ is closed if and only if S(ϕ) = {0}, hence, by the closed graph theorem ϕ is continuous if and only if S(ϕ) = {0}. The following first appeared in [DHK01] as an adaptation of a lemma by Sinclair, the origins of which can be traced back to [Sin76]. For every WPS (σ, w) we can define the weight function of the system to be w σ (x) = P (σ, w)(1)(x) = i:x∈X i w i (x) which is positive continuous function that vanishes only on X − ∪ d i=1 X i .…”

Section: Universal Description and Automatic Continuitymentioning

confidence: 97%

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Isomorphisms of tensor algebras arising from weighted partial systems

Dor-On

2018

Trans. Amer. Math. Soc.

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Abstract. 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.

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Section: Universal Description and Automatic Continuitymentioning

confidence: 97%

“…In Section 5 we show that tensor algebras arising from the C*-correspondence of a WPS possess a certain universal property by using tools from Section 3 of [Kwa14], and find conditions on the WPS to ensure any isomorphism onto the tensor algebra of a WPS is automatically continuous, by applying ideas from [DHK01].…”

Section: Introductionmentioning

confidence: 99%

Isomorphisms of tensor algebras arising from weighted partial systems

Dor-On

2018

Trans. Amer. Math. Soc.

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“…Under certain conditions, we will show that algebraic isomorphisms are automatically bounded. We will follow closely the ideas of Donsig, Hudson and Katsoulis [DHK01], Katsoulis and Kribs [KK04], and Davidson and Katsoulis [DK11].…”

Section: Tensor Algebras and Their Graded Structurementioning

confidence: 99%

“…It is easy to see that the graph of ϕ is closed if and only if S(ϕ) = {0}, hence by the closed graph theorem we have that φ is continuous if and only if S(ϕ) = {0}. We will use the following adaptation of [Sin75, Lemma 2.1], which appeared first in Donsig, Hudson and Katsoulis [DHK01].…”

Section: Tensor Algebras and Their Graded Structurementioning

confidence: 99%

Operator algebras and subproduct systems arising from stochastic matrices

Dor-On

Markiewicz

2014

Journal of Functional Analysis

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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 Viselter's 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 Ω, and let T+(X) and T+(Y ) be their tensor algebras. We show that every algebraic isomorphism from T+(X) onto T+(Y ) is automatically bounded. Furthermore, T+(X) and T+(Y ) are isometrically isomorphic if and only if X and Y are unitarily isomorphic up to a *-automorphism of ℓ ∞ (Ω). When Ω is finite, we prove that T+(X) and T+(Y ) are algebraically isomorphic if and only if there exists a similarity between X and Y up to a *-automorphism of ℓ ∞ (Ω). Moreover, we provide an explicit description of the Cuntz-Pimsner algebra O(X) in the case where Ω is finite and the stochastic matrix is essential.1 2 ADAM DOR-ON AND DANIEL MARKIEWICZ is easy to find two graphs with d vertices and irreducible adjacency matrix whose C*-algebras are not isomorphic (see [Rae05]). In contrast, in section 5 we show that if X is the Arveson-Stinespring subproduct system of a d × d irreducible stochastic matrix, then O(X) ∼ = C(T) ⊗ M d (C). More generally, we also provide an explicit description for the Cuntz-Pimsner algebra of a subproduct system associated to essential finite stochastic matrices.On the other hand, the non-self-adjoint tensor algebra T + (E) of a C*-correspondence E over A has often proven to be a strong invariant of the correspondence. Muhly and Solel [MS00] proved that if E and F are aperiodic C*-correspondences, then T + (E) is isometrically isomorphic to T + (F ) if and only if E and F are isometrically isomorphic as A-bimodules. Similarly, Katsoulis and Kribs [KK04] and Solel [Sol04] proved that if G and G ′ are countable directed graphs, then the tensor algebras T + (G) and T + (G ′ ) are isomorphic as algebras if and only if G and G ′ are isomorphic as directed graphs. See also Davidson and Katsoulis [DK11] for another important example of this phenomenon of increased acuity of the normed (non-self-adjoint) algebras as opposed to C*algebras, perhaps first recognized in Arveson [Arv67] and Arveson and Josephson [AJ69]. The tensor algebras of subproduct systems were first considered by Solel and Shalit [SS09] in the special case of M = C, and they analyzed the problem of graded isomorphism of their tensor algebras. The general isomorphism problem for such subproduct systems was resolved by Davidson, Ramsey and Shalit [DRS11]. They proved that if X, Y are subproduct systems of finitedimensional Hilbert space fibers, then T + (X) and T + (Y ) are isometrically isomorphic if and only if X and Y are (unitarily) isomorphic.On the other hand, the recent work of Gurevich [Gur12] provides a useful contrast. Although in this paper we focus on subproduct systems over N, it is possible to consider mo...

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“…Stephen Power [15] has used this fact to prove that the spectrum of an AF C*-algebra is a complete isomorphism invariant up to isometric isomorphism. But Donsig, Katsoulis and Hudson [5] have shown that isometric isomorphism is equal to algebraic isomorphism. For TAF algebras, since there is only one canonical masa, the spectrum is an invariant of the algebra.…”

Section: Letmentioning

confidence: 99%

Principal ideals in subalgebras of groupoid $C^*$-algebras

Krishnan¹

2002

Illinois J. Math.

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The study of different types of ideals in non self-adjoint operator algebras has been a topic of recent research. This study focuses on principal ideals in subalgebras of groupoid C*-algebras. An ideal is said to be principal if it is generated by a single element of the algebra. We look at subalgebras of r-discrete principal groupoid C*-algebras and prove that these algebras are principal ideal algebras. Regular canonical subalgebras of almost finite C*-algebras have digraph algebras as their building blocks. The spectrum of almost finite C*-algebras has the structure of an r-discrete principal groupoid and this helps in the coordinization of these algebras. Regular canonical subalgebras of almost finite C*-algebras have representations in terms of open subsets of the spectrum for the enveloping C*-algebra. We conclude that regular canonical subalgebras are principal ideal algebras.

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Algebraic isomorphisms of limit algebras

Cited by 11 publications

References 22 publications

Isomorphisms of tensor algebras arising from weighted partial systems

Isomorphisms of tensor algebras arising from weighted partial systems

Operator algebras and subproduct systems arising from stochastic matrices

Principal ideals in subalgebras of groupoid $C^*$-algebras

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