Two families of Exel–Larsen crossed products

Brownlowe, Nathan; Raeburn, Iain

doi:10.1016/j.jmaa.2012.08.026

Cited by 6 publications

(3 citation statements)

References 34 publications

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“…We also have an action of the opposite semigroup P op by unital, positive, linear maps L p : C * (G) → C * (G) given by L p (δ g ) = χ θp(G) (g)δ θ −1 p (g) . Each L p is a transfer operator for (C * (G), α p ), in the sense that L p (α p (a)b) = aL p (b) for all a, b ∈ C * (G), and so (C * (G), P, α, L) is a dynamical system in the style of the Exel-Larsen systems studied in [BR13,Lar10].…”

Section: Nica-toeplitz Algebras For Algebraic Dynamical Systemsmentioning

confidence: 99%

C^*-Algebras of algebraic dynamical systems and right LCM semigroups

Brownlowe¹,

Larsen²,

Stammeier³

2018

Indiana Univ. Math. J.

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We introduce algebraic dynamical systems, which consist of an action of a right LCM semigroup by injective endomorphisms of a group. To each algebraic dynamical system we associate a C * -algebra and describe it as a semigroup C * -algebra. As part of our analysis of these C * -algebras we prove results for right LCM semigroups. More precisely we discuss functoriality of the full semigroup C * -algebra and compute its K-theory for a large class of semigroups. We introduce the notion of a Nica-Toeplitz algebra of a product system over a right LCM semigroup, and show that it provides a useful alternative to study algebraic dynamical systems.

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Section: Nica-toeplitz Algebras For Algebraic Dynamical Systemsmentioning

confidence: 99%

C^*-Algebras of algebraic dynamical systems and right LCM semigroups

Brownlowe¹,

Larsen²,

Stammeier³

2018

Indiana Univ. Math. J.

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“…(The proof of this in [5] is algebraic: when |Λ 0 | = 1, the path space Λ is a semigroup, and one can study the graph by studying the algebraic properties of this semigroup. There is an alternative graph-based proof in the appendix to [3]. )…”

Section: This Gives (B)mentioning

confidence: 99%

Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

Huef

Kang

Raeburn

2015

Journal of Mathematical Analysis and Applications

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Several authors have recently been studying the equilibrium or KMS states on the Toeplitz algebras of finite higher-rank graphs. For graphs of rank one (that is, for ordinary directed graphs), there is a natural dynamics obtained by lifting the gauge action of the circle to an action of the real line. The algebras of higher-rank graphs carry a gauge action of a higher-dimensional torus, and there are many potential dynamics arising from different embeddings of the real line in the torus. Previous results show that there is nonetheless a "preferred dynamics" for which the system exhibits a particularly satisfactory phase transition, and that the unique KMS state at the critical inverse temperature can then be implemented by integrating vector states against a measure on the infinite path space of the graph. Here we obtain a similar description of the KMS state at the critical inverse temperature for other dynamics. Our spatial implementation is given by integrating against a measure on a space of paths which are infinite in some directions but finite in others. Our results are sharpest for the algebras of rank-two graphs.

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“…Our results are interesting already for the commutative Ore monoids (N k , +). Several authors have considered examples of product systems over these and other commutative cancellative monoids ( [10,31,32,51,79]). Commutativity seems to be a red herring: what is relevant are Ore conditions.…”

Section: Introductionmentioning

confidence: 99%

Product systems over Ore monoids

Albandik¹,

Meyer

2015

Doc. Math.

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We interpret the Cuntz-Pimsner covariance condition as a nondegeneracy condition for representations of product systems. We show that Cuntz-Pimsner algebras over Ore monoids are constructed through inductive limits and section algebras of Fell bundles over groups. We construct a groupoid model for the Cuntz-Pimsner algebra coming from an action of an Ore monoid on a space by topological correspondences. We characterise when this groupoid is effective or locally contracting and describe its invariant subsets and invariant measures.

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Two families of Exel–Larsen crossed products

Cited by 6 publications

References 34 publications

C^*-Algebras of algebraic dynamical systems and right LCM semigroups

C^*-Algebras of algebraic dynamical systems and right LCM semigroups

Spatial realisations of KMS states on theC⁎-algebras of higher-rank graphs

Product systems over Ore monoids

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