Uniqueness theorems for topological higher-rank graph 𝐶*-algebras

Renault, Jean; Sims, Aidan; Williams, Dana P.; Yeend, Trent

doi:10.1090/proc/13745

Cited by 10 publications

(12 citation statements)

References 27 publications

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“…Every UCT Kirchberg algebra has a Cartan subalgebra. That follows from [9, Thereom C] and [28,29], see also [21]. As every separable, nuclear C*-algebra is KK-equivalent to a Kirchberg algebra by [10, Theorem I], we therefore conclude that the UCT problem has a positive answer if and only if every Kirchberg algebra admits a Cartan subalgebra.…”

Section: Preliminariesmentioning

confidence: 57%

Cartan subalgebras and the UCT problem

Barlak

2017

Advances in Mathematics

101

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Abstract. We show that a separable, nuclear C*-algebra satisfies the UCT if it has a Cartan subalgebra. Furthermore, we prove that the UCT is closed under crossed products by group actions which respect Cartan subalgebras. This observation allows us to deduce, among other things, that a crossed product O 2 ⋊ α Z p satisfies the UCT if there is some automorphism γ of O 2 with the property that γ(D 2 ) ⊆ O 2 ⋊ α Z p is regular, where D 2 denotes the canonical masa of O 2 . We prove that this condition is automatic if γ(D 2 ) ⊆ O 2 ⋊ α Z p is not a masa or α(γ(D 2 )) is inner conjugate to γ(D 2 ). Finally, we relate the UCT problem for separable, nuclear, M 2 ∞ -absorbing C*-algebras to Cartan subalgebras and order two automorphisms of O 2 .

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Section: Preliminariesmentioning

confidence: 57%

Cartan subalgebras and the UCT problem

Barlak

2017

Advances in Mathematics

101

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“…Unlike [13] and [15], we do not require our graphs to be countable. More general versions are described in [9,12,18,26]. Define…”

Section: Preliminariesmentioning

confidence: 99%

Equivalent groupoids have Morita equivalent Steinberg algebras

Clark

Sims

2015

Journal of Pure and Applied Algebra

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Abstract. Let G and H be Hausdorff ample groupoids and let R be a commutative unital ring. We show that if G and H are equivalent in the sense of Muhly-RenaultWilliams, then the associated Steinberg algebras of locally constant R-valued functions with compact support are Morita equivalent. We deduce that collapsing a "collapsible subgraph" of a directed graph in the sense of Crisp and Gow does not change the Morita-equivalence class of the associated Leavitt path R-algebra, and therefore a number of graphical constructions which yield Morita equivalent C

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“…They are also very useful for studying structural properties like primeness and simplicity. Appropriate versions of these theorems have been proved not just for Leavitt path algebras but also (and we refer to [30,31,58,61]) for graph C * -algebras, as well as Cohn path algebras, higher-rank graph algebras, and even algebras of topological higher-rank graphs.…”

Section: 2mentioning

confidence: 99%

“…The construction was later generalised in a number of different directions, taking a route through inverse semigroup theory [57], and going as far as topological higher-rank graphs (e.g. [49,61,71]…”

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confidence: 99%

The Groupoid Approach to Leavitt Path Algebras

Rigby

2020

Leavitt Path Algebras and Classical K-Theory

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When the theory of Leavitt path algebras was already quite advanced, it was discovered that some of the more difficult questions were susceptible to a new approach using topological groupoids. The main result that makes this possible is that the Leavitt path algebra of a graph is graded isomorphic to the Steinberg algebra of the graph's boundary path groupoid.This expository paper has three parts: Part 1 is on the Steinberg algebra of a groupoid, Part 2 is on the path space and boundary path groupoid of a graph, and Part 3 is on the Leavitt path algebra of a graph. It is a self-contained reference on these topics, intended to be useful to beginners and experts alike. While revisiting the fundamentals, we prove some results in greater generality than can be found elsewhere, including the uniqueness theorems for Leavitt path algebras.

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Uniqueness theorems for topological higher-rank graph 𝐶*-algebras

Cited by 10 publications

References 27 publications

Cartan subalgebras and the UCT problem

Cartan subalgebras and the UCT problem

Equivalent groupoids have Morita equivalent Steinberg algebras

The Groupoid Approach to Leavitt Path Algebras

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