The C*-envelope of the Tensor Algebra of a Directed Graph

Katsoulis, Elias G.; Kribs, David W.

doi:10.1007/s00020-006-1430-8

Cited by 16 publications

(19 citation statements)

References 26 publications

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“…In fact, we have shown that Szymanski's Theorem is not necessary and that one can obtain Theorem 7.7 by using the more elementary "gauge-invariant uniqueness theorem" [21].…”

Section: Acknowledgementsmentioning

confidence: 98%

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Katsoulis

Kribs

2005

Trans. Amer. Math. Soc.

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Abstract. Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of operators on Hilbert space which are related to the graph in a natural way. This yields a functional calculus determined by quiver algebras and free semigroupoid algebras. We establish a generalization of Coburn's theorem for the C * -algebra of a CKT family, and prove a universality theorem for C * -algebras generated by these families. In both cases, the C * -algebras generated by quiver algebras play the universal role.

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“…In fact, we have shown that Szymanski's Theorem is not necessary and that one can obtain Theorem 7.7 by using the more elementary "gauge-invariant uniqueness theorem" [21].…”

Section: Acknowledgementsmentioning

confidence: 98%

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Katsoulis

Kribs

2005

Trans. Amer. Math. Soc.

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“…Consequently the associated Cuntz-Pimsner algebra O(E) is nuclear [26,Corollary 7.5]. By a result of Katsoulis and Kribs [23], O(E) is the C*-envelope of T + (E).…”

Section: Dilation and The Semi-crossed Productmentioning

confidence: 99%

Operator algebras for multivariable dynamics

2011

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Let X be a locally compact Hausdorff space with n proper continuous self maps σ i : X → X for 1 ≤ i ≤ n. To this we associate two conjugacy operator algebras which emerge as the natural candidates for the universal algebra of the system, the tensor algebra A(X, τ ) and the semicrossed product C 0 (X) × τ F + n . We develop the necessary dilation theory for both models. In particular, we exhibit an explicit family of boundary representations which determine the C*-envelope of the tensor algebra.We introduce a new concept of conjugacy for multidimensional systems, called piecewise conjugacy. We prove that the piecewise conjugacy class of the system can be recovered from the algebraic structure of either A(X, σ) or C 0 (X) × σ F + n . Various classification results follow as a consequence. For example, if n = 2 or 3, or the space X has covering dimension at most 1, then the tensor algebras are algebraically isomorphic (or completely isometrically isomorphic) if and only if the systems are piecewise topologically conjugate.We define a generalized notion of wandering sets and recurrence. Using this, it is shown that A(X, σ) or C 0 (X) × σ F + n is semisimple if and only if there are no generalized wandering sets. In the metrizable case, this is equivalent to each σ i being surjective and v-recurrent points being dense for each v ∈ F + n .

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“…We now obtain one of the main results of [7] as a corollary. Note that in [7], the proof of the above corollary is essentially self-contained and avoids the heavy machinery used in this paper.…”

Section: Lemma 32 If X Be a Faithful C * -Correspondence Over A Thenmentioning

confidence: 90%

“…In the special case of a graph correspondence, this was done in [7] with the help of a wellknown process called "adding tails to a graph." This process has been generalized to arbitrary correspondences by Muhly and Tomforde [12].…”

Section: Lemma 32 If X Be a Faithful C * -Correspondence Over A Thenmentioning

confidence: 99%

“…Our proof does not require any of the results from [11] and is modelled upon the proof of our recent result [7] that identifies the C * -envelope of the tensor algebra of a directed graph. We also make use of the result of Muhly and Tomforde [12] that generalizes the process of adding tails to a graph to the context of C * -correspondences.…”

Section: Introductionmentioning

confidence: 97%

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Tensor algebras ofC∗-correspondences and theirC∗-envelopes

Katsoulis

Kribs

2006

Journal of Functional Analysis

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We show that the C * -envelope of the tensor algebra of an arbitrary C * -correspondence X coincides with the Cuntz-Pimsner algebra O X , as defined by Katsura [T. Katsura, On C * -algebras associated with C * -correspondences, J. Funct. Anal. 217 (2004) 366-401]. This improves earlier results of Muhly and Solel [P.S. Muhly, B. Solel, Tensor algebras over C * -correspondences: Representations, dilations and C * -envelopes, J. Funct. Anal. 158 (1998) 389-457] and Fowler, Muhly and Raeburn [N. Fowler, P. Muhly, I. Raeburn, Representations of Cuntz-Pimsner algebras, Indiana Univ. Math. J. 52 (2003) 569-605], who came to the same conclusion under the additional hypothesis that X is strict and faithful.

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The C*-envelope of the Tensor Algebra of a Directed Graph

Cited by 16 publications

References 26 publications

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Applications of the Wold decomposition to the study of row contractions associated with directed graphs

Operator algebras for multivariable dynamics

Tensor algebras ofC∗-correspondences and theirC∗-envelopes

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