scholarly journals Exel's crossed product for non-unital C*-algebras

2010 ◽  
Vol 149 (3) ◽  
pp. 423-444 ◽  
Author(s):  
NATHAN BROWNLOWE ◽  
IAIN RAEBURN ◽  
SEAN T. VITTADELLO
Keyword(s):  
Dynamical Systems ◽  
Transfer Operator ◽  
Finite Graph ◽  
Crossed Product ◽  
Crossed Products ◽  
Ideal Structure ◽  
Locally Finite ◽  
C Algebras ◽  
Locally Finite Graph ◽  
The Ideal

AbstractWe consider a family of dynamical systems (A, α, L) in which α is an endomorphism of a C*-algebra A and L is a transfer operator for α. We extend Exel's construction of a crossed product to cover non-unital algebras A, and show that the C*-algebra of a locally finite graph can be realised as one of these crossed products. When A is commutative, we find criteria for the simplicity of the crossed product, and analyse the ideal structure of the crossed product.

scholarly journals Ideal structure of crossed products by endomorphisms via reversible extensions of C*-dynamical systems

2015 ◽  
Vol 26 (03) ◽  
pp. 1550022 ◽  
Author(s):  
Bartosz Kosma Kwaśniewski
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Transfer Operator ◽  
Crossed Product ◽  
Crossed Products ◽  
Ideal Structure ◽  
Gauge Invariant ◽  
C Algebra ◽  
System A ◽  
The Ideal

We consider an extendible endomorphism α of a C*-algebra A. We associate to it a canonical C*-dynamical system (B, β) that extends (A, α) and is "reversible" in the sense that the endomorphism β admits a unique regular transfer operator β⁎. The theory for (B, β) is analogous to the theory of classic crossed products by automorphisms, and the key idea is to describe the counterparts of classic notions for (B, β) in terms of the initial system (A, α). We apply this idea to study the ideal structure of a non-unital version of the crossed product C*(A, α, J) introduced recently by the author and A. V. Lebedev. This crossed product depends on the choice of an ideal J in (ker α)⊥, and if J = ( ker α)⊥ it is a modification of Stacey's crossed product that works well with non-injective α's. We provide descriptions of the lattices of ideals in C*(A, α, J) consisting of gauge-invariant ideals and ideals generated by their intersection with A. We investigate conditions under which these lattices coincide with the set of all ideals in C*(A, α, J). In particular, we obtain simplicity criteria that besides minimality of the action require either outerness of powers of α or pointwise quasinilpotence of α.

scholarly journals Rokhlin dimension: duality, tracial properties, and crossed products

2019 ◽  
Vol 41 (2) ◽  
pp. 408-460
Author(s):  
EUSEBIO GARDELLA ◽  
ILAN HIRSHBERG ◽  
LUIS SANTIAGO
Keyword(s):  
Compact Group ◽  
Local Approximation ◽  
Stable Rank ◽  
Crossed Product ◽  
Crossed Products ◽  
Real Rank ◽  
Ideal Structure ◽  
The Core ◽  
Universal Coefficient Theorem ◽  
The Ideal

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$-algebras, including: $D$-absorbing $C^{\ast }$-algebras, where $D$ is a strongly self-absorbing $C^{\ast }$-algebra; stable $C^{\ast }$-algebras; $C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$-algebras with finite stable rank (or real rank); and $C^{\ast }$-algebras whose $K$-theory is either trivial, rational, or $n$-divisible for $n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

scholarly journals Topologically free actions and ideals in discrete C*-dynamical systems

1994 ◽  
Vol 37 (1) ◽  
pp. 119-124 ◽  
Author(s):  
R. J. Archbold ◽  
J. S. Spielberg
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Natural Condition ◽  
Discrete System ◽  
Crossed Product ◽  
Ideal Structure ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
Free Actions ◽  
The Ideal

A C*-dynamical system is called topologically free if the action satisfies a certain natural condition weaker than freeness. It is shown that if a discrete system is topologically free then the ideal structure of the crossed product algebra is related to that of the original algebra. One consequence is that a minimal topologically free discrete system has a simple reduced crossed product. Sharper results are obtained when the algebra is abelian.

Asymptotic Behavior of Skew Conditional Heat Kernels on Graph Networks

1993 ◽  
Vol 45 (4) ◽  
pp. 863-878 ◽  
Author(s):  
Tatsuya Okada
Keyword(s):  
Asymptotic Behavior ◽  
Finite Graph ◽  
Heat Propagation ◽  
Heat Kernels ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Periodic Models ◽  
Definition Of

AbstractIn this note, we will consider the heat propagation on locally finite graph networks which satisfy a skew condition on vertices (See Definition of Section 2). For several periodic models, we will construct the heat kernels Pt with the skew condition explicitly, and derive the decay order of Pt as time goes to infinity.

scholarly journals Extension problems and non-Abelian duality for C*-algebras

2007 ◽  
Vol 75 (2) ◽  
pp. 229-238 ◽  
Author(s):  
Astrid an Huef ◽  
S. Kaliszewski ◽  
Iain Raeburn
Keyword(s):  
Compact Group ◽  
Unitary Representation ◽  
Closed Subgroup ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Test Question ◽  
Locally Compact ◽  
Dual Representation ◽  
C Algebras

Suppose that H is a closed subgroup of a locally compact group G. We show that a unitary representation U of H is the restriction of a unitary representation of G if and only if a dual representation Û of a crossed product C*(G) ⋊ (G/H) is regular in an appropriate sense. We then discuss the problem of deciding whether a given representation is regular; we believe that this problem will prove to be an interesting test question in non-Abelian duality for crossed products of C*-algebras.

scholarly journals The homology of a locally finite graph with ends

COMBINATORICA ◽  
2010 ◽  
Vol 30 (6) ◽  
pp. 681-714 ◽  
Author(s):  
Reinhard Diestel ◽  
Philipp Sprüssel
Keyword(s):  
Finite Graph ◽  
Locally Finite ◽  
Locally Finite Graph

A CONSTRUCTION FOR WEIGHTS ON C*-ALGEBRAS: DUAL WEIGHTS FOR C*-CROSSED PRODUCTS

1999 ◽  
Vol 10 (01) ◽  
pp. 129-157 ◽  
Author(s):  
J. QUAEGEBEUR ◽  
J. VERDING
Keyword(s):  
Dynamical System ◽  
Haar Measure ◽  
Crossed Product ◽  
Crossed Products ◽  
C Algebras ◽  
System A ◽  
Lower Semi ◽  
Continuous Weight ◽  
Densely Defined

A method for constructing densely defined lower semi-continuous weights on C*-algebras is presented. The method can be used to construct a "dual weight" on the C*-crossed product A×αG of a C*-dynamical system (A,G,α), starting from a relatively α-invariant densely defined lower semi-continuous weight on A. As an application we show that the Haar measure on the quantum E(2) group is a C*-dual weight.

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