Group actions on topological graphs

Deaconu, Valentin; Kumjian, Alex; Quigg, John

doi:10.1017/s014338571100040x

Cited by 21 publications

(27 citation statements)

References 28 publications

(25 reference statements)

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“…We will need to know how the homomorphism Π from [1] can be described using the techniques of [7]: [1, Proof of Theorem 5.6] constructs a correspondence homomorphism (ψ, π) : (X, A) → (M B (Y ), M(B)), although the notation in [1] is substantially different 3…”

Section: A Direct Approach To the Coactionmentioning

confidence: 99%

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Coactions and Skew Products of Topological Graphs

Kaliszewski

Quigg

2012

Integr. Equ. Oper. Theory

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Section: A Direct Approach To the Coactionmentioning

confidence: 99%

“…The roles of E, X, A and F, Y, B are interchanged, and what we call (ψ, π) here was written as (µ, ν) in[1].…”

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confidence: 99%

Coactions and Skew Products of Topological Graphs

Kaliszewski

Quigg

2012

Integr. Equ. Oper. Theory

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“…In this section we extend the notions of [DKQ12, section 3] with the aim of showing that the only obstructions to a topological quiver being a skew product are the existence of a free and proper action by G and a global section for the quotient. We remark that [DKQ12] goes considerably further in their analysis of group actions on topological graphs, which we leave to pursue in future work.…”

Section: Preliminariesmentioning

confidence: 98%

Coactions and Skew Products for Topological Quivers

Hall¹

2021

Preprint

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Given a cocycle on a topological quiver by a locally compact group, the author constructs a skew product topological quiver, and determines conditions under which a topological quiver can be identified as a skew product. We investigate the relationship between the C * -algebra of the skew product and a certain native coaction on the C * -algebra of the original quiver, finding that the crossed product by the coaction is isomorphic to the skew product. As an application, we show that the reduced crossed product by the dual action is Morita equivalent to the C * -algebra of the original quiver. HALLEquipped with the technology of Cuntz-Pimsner covariant homomorphisms into C * -algebras, Kaliszewski, Quigg, and Robertson introduced the notion of Cuntz-Pimsner covariance for morphisms between correspondences [KQR13]. There, they showed that the passage from C * -correspondences to the Cuntz-Pimsner algebra was functorial between a suitible pair of categories. Shortly after, these three authors investigated this functor for correspondences equipped with a coaction[KQR15], deducing that a suitable correspondence coaction generates a coaction on the associated Cuntz-Pimsner algebra. Kaliszewski and Quigg then used this machinery to recover for Katsura's topological graphs what had been discovered for directed graphs [KQ13]. The skew product topological graph by a (locally compact) group is isomorphic to the crossed product by a naturally occuring coaction, producing this time a coaction that you can "see" in the continuous setting.Around the time that Katsura studied his topological graphs, Muhly and Solel introduced topological quivers to exemplify traits in the study of Morita equivalence of tensor algebras. Muhly and Tomforde later took up the task of investigating the associated C * -algebras in [MT05]. Apart from the matter of convention (and we conform to the conventions of Katsura here), the principle difference between quivers and topological graphs is that topological graphs insist that the source map is a local homeomorphism, while quivers insist only that the source map is open. To make up for this freedom, we insist that the fibres over any vertex can be appropriately measured, and the authors follow the strategy used for measuring groupoids by a system of measures. In this article, we take up the study of skew products for topological quivers.In §2, we record our conventions for bundles, topological quivers and associated constructions, and coactions. Starting with §3 we introduce the skew product quiver and produce a Gross-Tucker type classification for skew products. In §4 we study the associated C * -algebras in earnest, culminating in the main objective Theorem 4.3. We end with a section on applications, where we offer an open problem regarding qualities of the coaction, and show that the reduced crossed product by the dual action is Morita equivalent to the C * -algebra of the original quiver.

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“…The main purpose of relative multipliers is the following extension theorem [DKQar,Proposition A.11]: let X and Y be nondegenerate correspondences over A and B, respectively, let κ : C → M(A) and σ : D → M(B) be nondegenerate homomorphisms. If there is a nondegenerate homomorphism λ : C → M(σ(D)) such that π(κ(c)a) = λ(c)π(a) for c ∈ C, a ∈ A, then for any correspondence homomorphism (ψ, π) : (X, A) → (M D (Y ), M D (B)) there is a unique C-strict to D-strictly continuous correspondence homomorphism (ψ, π) making the diagram (X, A)…”

Section: Preliminariesmentioning

confidence: 99%