Simplicity of algebras associated to étale groupoids

“…With all of this theory in place, along with the simplicity results of [23] and [4], the following is an immediate corollary of Theorem 4.1 and Corollary 4.2.…”

Section: Topologically Principal Groupoids and Positive Elements Ofmentioning

confidence: 86%

“…Recall that a Kirchberg algebra is a separable, nuclear, purely infinite simple C * -algebra. We combine Theorem 3.3 with results from [2,4,21] to obtain the following characterization of groupoid Kirchberg algebras. …”

Section: Topologically Principal Groupoids and Positive Elements Ofmentioning

confidence: 99%

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Purely infinite -algebras associated to étale groupoids

Brown¹,

Clark²,

Sierakowski³

2014

Ergod. Th. Dynam. Sys.

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Abstract. Let G be a Hausdorff,étale groupoid that is minimal and topologically principal. We show that C * r (G) is purely infinite simple if and only if all the nonzero positive elements of C 0 (G (0) ) are infinite in C * r (G). If G is a Hausdorff, ample groupoid, then we show that C * r (G) is purely infinite simple if and only if every nonzero projection in C 0 (G (0) ) is infinite in C * r (G). We then show how this result applies to k-graph C * -algebras. Finally, we investigate strongly purely infinite groupoid C * -algebras.

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“…Our definition of effectiveness is from Lemma 3.1 of [11]. That lemma gives several equivalent characterisations of effective.…”

Section: Groupoids and Steinberg Algebrasmentioning

confidence: 99%

Cohn Path Algebras of Higher-Rank Graphs

Clark

Pangalela

2016

Algebr Represent Theor

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Abstract. In this article, we introduce Cohn path algebras of higher-rank graphs. We prove that for a higher-rank graph Λ, there exists a higher-rank graph T Λ such that the Cohn path algebra of Λ is isomorphic to the Kumjian-Pask algebra of T Λ. We then use this isomorphism and properties of Kumjian-Pask algebras to study Cohn path algebras. This includes proving a uniqueness theorem for Cohn path algebras. IntroductionLeavitt path algebras were introduced and studied in [3] and [9] as a generalisation the class of algebras studied by Leavitt in [25]. Leavitt path algebras are also the natural algebraic analogues of graph C * -algebras studied in [32]; a number of interesting result have been proven by translating between graph C * -algebras and the ring theoretic Leavitt path algebras.Cohn path algebras were introduced in [6,7] and generalise the algebras U 1,n studied by Cohn in [16]. The idea is to build an algebra out of the path space of a graph; addition and scalar multiplication are defined formally and multiplication of two paths is only nonzero when one can concatenate the paths. Cohn path algebras can also be obtained from Leavitt path algebras by omitting one of the Cuntz-Krieger relations. Hence every Leavitt path algebra can be viewed as a quotient of a Cohn path algebra. On the other hand, Abrams, Ara and Siles Molina show in [2] that for every graph E, there exists a graph T E (denoted E (X) in [2]) such that the Cohn path algebra of E is isomorphic to the Leavitt path algebra of T E.Although it has received less attention, the Cohn path algebra of a directed graph is the algebraic analogue of the C * -algebraic Toeplitz algebra of E as defined in [19]. In the C * -algebraic setting, Muhly and Tomforde in [27] and also Sims in [38] each show that for a graph E, the Toeplitz algebra of E is isomorphic to the graph C * -algebra of T E. Now we move into the setting of higher-rank graph algebras: In [24], Kumjian and Pask introduced a combinatorial model, called a higher-rank graph, in order to capture the essential features of the C * -algebras studied by Robertson and Steger in [35]. A higherrank graph, also called a k-graph, is a generalisation of the path category of a directed graph where the length of a 'path' λ in a k-graph is an element of N k . Kumjian and Pask studied the C * -algebras associated to row-finite higher-rank graphs with no sources. Raeburn, Sims and Yeend generalised Kumjian and Pask's construction by describing the class of C * -algebras associated to more general higher-rank graphs in [30,31].1991 Mathematics Subject Classification. 16S99 (Primary); 16S10 (Secondary).

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Simplicity of algebras associated to étale groupoids

Cited by 126 publications

References 30 publications

Uniqueness Theorems for Steinberg Algebras

Uniqueness Theorems for Steinberg Algebras

Purely infinite -algebras associated to étale groupoids

Cohn Path Algebras of Higher-Rank Graphs

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