Groupoid Fell bundles for product systems over quasi-lattice ordered groups

Rennie, Adam; Robertson, David; Sims, Aidan

doi:10.1017/s0305004117000202

Cited by 6 publications

(9 citation statements)

References 21 publications

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“…The previous proposition combined with [27,Theorem 6.3] gives us the following: Example 4.5. Let F 2 denote the free group on two generators a and b.…”

Section: Relationship To Other Constructionsmentioning

confidence: 99%

On C⁎-algebras associated to product systems

Sehnem

2019

Journal of Functional Analysis

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Let P be a unital subsemigroup of a group G. We propose an approach to C * -algebras associated to product systems over P . We call the C * -algebra of a given product system E its covariance algebra and denote it by A× E P , where A is the coefficient C * -algebra. We prove that our construction does not depend on the embedding P → G and that a representation of A × E P is faithful on the fixed-point algebra for the canonical coaction of G if and only if it is faithful on A. We compare this with other constructions in the setting of irreversible dynamical systems, such as Cuntz-Nica-Pimsner algebras, Fowler's Cuntz-Pimsner algebra, semigroup C * -algebras of Xin Li and Exel's crossed products by interaction groups.

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“…The previous proposition combined with [27,Theorem 6.3] gives us the following: Example 4.5. Let F 2 denote the free group on two generators a and b.…”

Section: Relationship To Other Constructionsmentioning

confidence: 99%

On C⁎-algebras associated to product systems

Sehnem

2019

Journal of Functional Analysis

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“…We just prove exactness of the first diagram: the second follows from a similar argument. Since A is nuclear, so is T X (see, for example, [34,Theorem 6.3]) and so the quotient map q : T X → O X has a completely positive splitting. Hence [39, Theorem 1.1] applied to the graded short exact sequence 0 → K(F X )…”

Section: Pimsner's Exact Sequences For Graded C * -Algebrasmentioning

confidence: 99%

Twisted topological graph algebras are twisted groupoid$C^∗$-algebras

Kumjian¹,

Li²

2017

J. Operator Theory

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We develop methods for computing graded K-theory of C * -algebras as defined in terms of Kasparov theory. We establish graded versions of Pimsner's six-term sequences for graded Hilbert bimodules whose left action is injective and by compacts, and a graded Pimsner-Voiculescu sequence. We introduce the notion of a twisted Pgraph C * -algebra and establish connections with graded C * -algebras. Specifically, we show how a functor from a P -graph into the group of order two determines a grading of the associated C * -algebra. We apply our graded version of Pimsner's exact sequence to compute the graded K-theory of a graph C * -algebra carrying such a grading.

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“…The expectation is that each one is (and hence both are) exact if and only if so is the diagonal A. This has been proven by several authors for general amenable quasi-lattices, such as Alabandik and Meyer [1], Rennie, Robertson and Sims [25] and Fletcher [10]. Nuclearity behaves in a similar manner for N T (X ) but not for N O(X ), leaving open the following question.…”

Section: Introductionmentioning

confidence: 94%

“…It has been verified that nuclearity of A implies that of N O(X ) in several cases, such as higher-rank graphs [17,18], C*-dynamics [14] and in other more general amenable contexts [1,25,26]. However, the converse is not true even at the level of Z + .…”

Section: Introductionmentioning

confidence: 99%

Finite-dimensional approximations for Nica–Pimsner algebras

Kakariadis¹

2019

Ergod. Th. Dynam. Sys.

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We give necessary and sufficient conditions for nuclearity of Cuntz–Nica–Pimsner algebras for a variety of quasi-lattice ordered groups. First we deal with the free abelian lattice case. We use this as a stepping-stone to tackle product systems over quasi-lattices that are controlled by the free abelian lattice and satisfy a minimality property. Our setting accommodates examples like the Baumslag–Solitar lattice for $n=m>0$ and the right-angled Artin groups. More generally, the class of quasi-lattices for which our results apply is closed under taking semi-direct and graph products. In the process we accomplish more. Our arguments tackle Nica–Pimsner algebras that admit a faithful conditional expectation on a small fixed point algebra and a faithful copy of the coefficient algebra. This is the case for CNP-relative quotients in-between the Toeplitz–Nica–Pimsner algebra and the Cuntz–Nica–Pimsner algebra. We complete this study with the relevant results on exactness.

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Groupoid Fell bundles for product systems over quasi-lattice ordered groups

Cited by 6 publications

References 21 publications

On C⁎-algebras associated to product systems

On C⁎-algebras associated to product systems

Twisted topological graph algebras are twisted groupoid$C^∗$-algebras

Finite-dimensional approximations for Nica–Pimsner algebras

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