Classification of Semicrossed Products of Finite-Dimensional C ∗ - Algebras

Dealba, Luz M.; Peters, J.A.F.

doi:10.2307/2045843

Cited by 6 publications

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“…, n − 1. The cycle algebra L Cn may be identified with the wot-closed semicrossed product C n × σ β Z + associated with the cyclic shift automorphism β of C n [13]. To see this identify L x i H G with H 2 for each i in the natural way (respecting word length).…”

Section: Free Semigroupoid Algebrasmentioning

confidence: 99%

“…This in turn is identifiable with the crossed product above. In fact, this matrix function algebra is the wot-closed variant of the matrix function algebra B n of De Alba and Peters [13] for the norm closed semicrossed product C n × σ β Z + . Such identifications are the Toeplitz versions of the identification of the graph C * -algebra of C n with M n (C(T)).…”

Section: Free Semigroupoid Algebrasmentioning

confidence: 99%

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Partly Free Algebras From Directed Graphs

Kribs

Power

2004

Current Trends in Operator Theory and Its Applications

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We say that a nonselfadjoint operator algebra is partly free if it contains a free semigroup algebra. Motivation for such algebras occurs in the setting of what we call free semigroupoid algebras. These are the weak operator topology closed algebras generated by the left regular representations of semigroupoids associated with finite or countable directed graphs. We expand our analysis of partly free algebras from previous work and obtain a graph-theoretic characterization of when a free semigroupoid algebra with countable graph is partly free. This analysis carries over to norm closed quiver algebras. We also discuss new examples for the countable graph case.Every finite or countable directed graph G recursively generates a Fock space Hilbert space and a family of partial isometries. These operators are of Cuntz-Krieger-Toeplitz type and also arise through the left regular representations of free semigroupoids determined by directed graphs. This was initially discovered by Muhly [23], and, in the case of finite graphs, more recent work with Solel [24] considered the norm closed algebras generated by these representations, which they called quiver algebras. In [19], we developed a structure theory for the weak operator topology closed algebras L G generated by the left regular representations coming from both finite and countable directed graphs; we called these algebras free semigroupoid algebras. In doing so, we found a unifying framework for a number of classes of algebras which appear in the literature, including; noncommutative analytic Toeplitz algebras L n [2,10,11,20,25,26] (the prototypical free semigroup algebras), the classical analytic Toeplitz algebra H ∞ [14,16], and certain finite dimensional digraph algebras [19]. But this approach gives rise to a diverse collection of new examples which include finite dimensional algebras, algebras with free behaviour, algebras which can be represented as matrix function algebras, and examples which mix these possibilities. The general theme of our work in [19] was a marriage of simple graph-theoretic properties with properties of the operator algebra. Furthermore, our technical analyses were chiefly spatial in nature; for instance, we proved the graph is a complete unitary invariant of both the free semigroupoid algebra and the quiver algebra.In the next section we give a short introduction to free semigroupoid algebras and discuss a number of examples. The second section contains an expanded analysis of a first author partially supported by a Canadian NSERC Post-doctoral Fellowship.

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Section: Free Semigroupoid Algebrasmentioning

confidence: 99%

Section: Free Semigroupoid Algebrasmentioning

confidence: 99%

Partly Free Algebras From Directed Graphs

Kribs

Power

2004

Current Trends in Operator Theory and Its Applications

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“…This and related work has attracted the interest of operator theorists over the last 45 years. For example, see [2,11,19,20,39,44,53,66,67,70,71,72,76,79,85,89,91]. When the semigroup is F ǹ with n ą 1, and the t i form a row contraction, then the resulting operator algebra is the tensor algebra related to a C*-correspondence.…”

Section: Introductionmentioning

confidence: 99%

Semicrossed products of operator algebras by semigroups

2017

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“…In the case where A is a C * -algebra and α preserves adjoints, all three choices produce the same operator algebra, Peters' semicrossed product of a C * -algebra [25] by an endomorphism. (Semicrossed products of C * -algebras have been under investigation by various authors [1,2,4,6,7,8,14,22,26], starting with the work of Arveson [2] in the late sixties.) In the general (non-selfadjoint) case however, the semicrossed products we introduce here may lead to non-isomorphic operator algebras.…”

Section: Introductionmentioning

confidence: 99%

Semicrossed products of C-algebras and their C-envelopes

Kakariadis

2016

JAMA

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Let A be a unital operator algebra and let α be an automorphism of A that extends to a * -automorphism of its C *envelope C * env (A). In this paper we introduce the isometric semicrossed product A × is α Z + and we show thatIn contrast, the C * -envelope of the familiar contractive semicrossed product A × α Z + may not equal C * env (A) × α Z. Our main tool for calculating C * -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 the main result of [9] to tensor algebras of C * -correspondences. We show that if T + X is the tensor algebra of a C * -correspondence (X , A) and α a completely isometric automorphism of T + X that fixes the diagonal elementwise, then the contractive semicrossed product satisfies C *where O X denotes the Cuntz-Pimsner algebra of (X , A).2000 Mathematics Subject Classification. 47L55, 47L40, 46L05, 37B20.

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Classification of Semicrossed Products of Finite-Dimensional C ∗ - Algebras

Cited by 6 publications

References 0 publications

Partly Free Algebras From Directed Graphs

Partly Free Algebras From Directed Graphs

Semicrossed products of operator algebras by semigroups

Semicrossed products of C-algebras and their C-envelopes

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Classification of Semicrossed Products of Finite-Dimensional C ∗ - Algebras

Cited by 6 publications

References 0 publications

Partly Free Algebras From Directed Graphs

Partly Free Algebras From Directed Graphs

Semicrossed products of operator algebras by semigroups

Semicrossed products of C*-algebras and their C*-envelopes

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Semicrossed products of C-algebras and their C-envelopes