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.

Ideal Structure of Multiplier Algebras of Simple C*-algebras With Real Rank Zero

2001 ◽  
Vol 53 (3) ◽  
pp. 592-630 ◽  
Author(s):  
Francesc Perera
Keyword(s):  
Stable Rank ◽  
Real Rank ◽  
Ideal Structure ◽  
Real Rank Zero ◽  
Von Neumann ◽  
Rank Zero ◽  
Rank One ◽  
A Chain ◽  
Multiplier Algebras ◽  
The Ideal

AbstractWe give a description of the monoid of Murray-von Neumann equivalence classes of projections for multiplier algebras of a wide class of σ-unital simple C*-algebras A with real rank zero and stable rank one. The lattice of ideals of this monoid, which is known to be crucial for understanding the ideal structure of themultiplier algebra , is therefore analyzed. In important cases it is shown that, if A has finite scale then the quotient of modulo any closed ideal I that properly contains A has stable rank one. The intricacy of the ideal structure of is reflected in the fact that can have uncountably many different quotients, each one having uncountably many closed ideals forming a chain with respect to inclusion.

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 C*-algebras of a Cantor system with finitely many minimal subsets: structures, K-theories, and the index map

2020 ◽  
pp. 1-46 ◽  
Author(s):  
SERGEY BEZUGLYI ◽  
ZHUANG NIU ◽  
WEI SUN
Keyword(s):  
Directed Graphs ◽  
Cantor Set ◽  
Stable Rank ◽  
Crossed Product ◽  
Real Rank ◽  
C Algebra ◽  
C Algebras ◽  
Rank 2 ◽  
Stably Finite ◽  
The Ideal

We study homeomorphisms of a Cantor set with $k$ ( $k<+\infty$ ) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$ , the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

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.

Spectra for Infinite Tensor Product Type Actions of Compact Groups

1996 ◽  
Vol 48 (2) ◽  
pp. 330-342
Author(s):  
Elliot C. Gootman ◽  
Aldo J. Lazar
Keyword(s):  
Tensor Product ◽  
Compact Group ◽  
Abelian Groups ◽  
Product Type ◽  
Crossed Product ◽  
Compact Groups ◽  
Ideal Structure ◽  
Crossed Product Algebra ◽  
Product Algebra ◽  
The Ideal

AbstractWe present explicit calculations of the Arveson spectrum, the strong Arveson spectrum, the Connes spectrum, and the strong Connes spectrum, for an infinite tensor product type action of a compact group. Using these calculations and earlier results (of the authors and C. Peligrad) relating the various spectra to the ideal structure of the crossed product algebra, we prove that the topology of G influences the ideal structure of the crossed product algebra, in the following sense: if G contains a nontrivial connected group as a direct summand, then the crossed product algebra may be prime, but it is never simple; while if G is discrete, the crossed product algebra is simple if and only if it is prime. These results extend to compact groups analogous results of Bratteli for abelian groups. In addition, we exhibit a class of examples illustrating that for compact groups, unlike the case for abelian groups, the Connes spectrum and strong Connes spectrum need not be stable.

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 Almost Finiteness for General Étale Groupoids and Its Applications to Stable Rank of Crossed Products

2018 ◽  
Vol 2020 (19) ◽  
pp. 6007-6041 ◽  
Author(s):  
Yuhei Suzuki
Keyword(s):  
Amenable Group ◽  
Cantor Set ◽  
Stable Rank ◽  
Crossed Product ◽  
Local Version ◽  
Crossed Products ◽  
Subexponential Growth ◽  
Minimal Action ◽  
New Strategy ◽  
Totally Disconnected

Abstract We extend Matui’s notion of almost finiteness to general étale groupoids and show that the reduced groupoid C$^{\ast }$-algebras of minimal almost finite groupoids have stable rank 1. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result: (1) for any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank 1; (2) any countable amenable group admits a minimal action on the Cantor set, all whose minimal extensions form the crossed product of stable rank 1; and (3) for any amenable group, the crossed product of the universal minimal action has stable rank 1.

scholarly journals Crossed products of Hilbert pro-C-bimodules and associated pro-C-algebras

2016 ◽  
Vol 32 (2) ◽  
pp. 195-201
Author(s):  
MARIA JOITA ◽  
◽  
RADU-B. MUNTEANU ◽  
Keyword(s):  
Compact Group ◽  
Inverse Limit ◽  
Locally Compact Group ◽  
Crossed Product ◽  
Crossed Products ◽  
Locally Compact ◽  
C Algebras

An action (γ, α) of a locally compact group G on a Hilbert pro-C∗-bimodule (X, A) induces an action γ × α of G on A ×X Z the crossed product of A by X. We show that if (γ, α) is an inverse limit action, then the crossed product of A ×α G by X ×γ G respectively of A ×α,r G by X ×γ,r G is isomorphic to the full crossed product of A ×X Z by γ × α respectively the reduced crossed product of A ×X Z by γ × α.

scholarly journals Equivariant KK-Theory and the Continuous Rokhlin Property

2020 ◽  
Author(s):  
Eusebio Gardella
Keyword(s):  
Fixed Point ◽  
Crossed Product ◽  
Compact Groups ◽  
Crossed Products ◽  
Circle Actions ◽  
Technical Result ◽  
C Algebras ◽  
Equivariant Version ◽  
Universal Coefficient Theorem

Abstract We introduce and study the continuous Rokhlin property for actions of compact groups on $C^*$-algebras. An important technical result is a characterization of the continuous Rokhlin property in terms of asymptotic retracts. As a consequence, we derive strong $KK$-theoretical obstructions to the continuous Rokhlin property. Using these, we show that the Universal Coefficient Theorem (UCT) is preserved under formation of crossed products and passage to fixed point algebras by such actions, even in the absence of nuclearity. As an application of the case of ${{\mathbb{Z}}}_3$-actions, we answer a question of Phillips–Viola about algebras not isomorphic to their opposites. Our analysis of the $KK$-theory of the crossed product allows us to prove a ${{\mathbb{T}}}$-equivariant version of Kirchberg–Phillips: two circle actions with the continuous Rokhlin property on Kirchberg algebras are conjugate whenever they are $KK^{{{\mathbb{T}}}}$-equivalent. In the presence of the UCT, this is equivalent to having isomorphic equivariant $K$-theory. We moreover characterize the $KK^{{{\mathbb{T}}}}$-theoretical invariants that arise in this way. Finally, we identify a $KK^{{{\mathbb{T}}}}$-theoretic obstruction to the continuous property, which is shown to be the only obstruction in the setting of Kirchberg algebras. We show by means of explicit examples that the Rokhlin property is strictly weaker than the continuous Rokhlin property.

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