scholarly journals An Equivariant Theory for the Bivariant Cuntz Semigroup

2018 ◽  
Vol 61 (2) ◽  
pp. 573-598
Author(s):  
Gabriele N. Tornetta
Keyword(s):  
Compact Group ◽  
Group Actions ◽  
Crossed Products ◽  
C Algebras ◽  
Regularity Properties ◽  
Cuntz Semigroup ◽  
The One ◽  
Equivariant Extension ◽  
Equivariant Theory

AbstractWe provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored, and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the one recently proposed by Gardella [‘Regularity properties and Rokhlin dimension for compact group actions’, Houston J. Math.43(3) (2017), 861–889], and we complement their work by providing an open projection picture of it.

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 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 Discrete coactions on C*-algebras

1996 ◽  
Vol 60 (1) ◽  
pp. 118-127 ◽  
Author(s):  
Chi-Keung Ng
Keyword(s):  
Fixed Point ◽  
Compact Group ◽  
Amenable Group ◽  
Group Actions ◽  
Crossed Product ◽  
Discrete Groups ◽  
C Algebras ◽  
Fixed Point Algebra

AbstractWe will consider coactions of discrete groups on C*-algebras and imitate some of the results about compact group actions on C*-algebras. In particular, the crossed product of a reduced coaction ∈ of a discrete amenable group G on A is liminal (respectively, postliminal) if and only if the fixed point algebra of ∈ is. Moreover, we will also consider ergodic coactions on C*-algebras.

MAXIMAL COACTIONS

2004 ◽  
Vol 15 (01) ◽  
pp. 47-61 ◽  
Author(s):  
SIEGFRIED ECHTERHOFF ◽  
S. KALISZEWSKI ◽  
JOHN QUIGG
Keyword(s):  
Compact Group ◽  
Group Actions ◽  
Locally Compact Group ◽  
Crossed Products ◽  
Locally Compact ◽  
Cross Sectional ◽  
C Algebra

A coaction δ of a locally compact group G on a C*-algebra A is maximal if a certain natural map from [Formula: see text] onto [Formula: see text] is an isomorphism. All dual coactions on full crossed products by group actions are maximal; a discrete coaction is maximal if and only if A is the full cross-sectional algebra of the corresponding Fell bundle. For every nondegenerate coaction of G on A, there is a maximal coaction of G on an extension of A such that the quotient map induces an isomorphism of the crossed products.

Crossed products by actions of finite groups with the Rokhlin property

2015 ◽  
Vol 26 (07) ◽  
pp. 1550042 ◽  
Author(s):  
Luis Santiago
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Group Actions ◽  
Crossed Product ◽  
Crossed Products ◽  
Finite Group Actions ◽  
C Algebras ◽  
Type Property ◽  
The Given

We introduce and study a Rokhlin-type property for actions of finite groups on (not necessarily unital) C*-algebras. We show that the corresponding crossed product C*-algebras can be locally approximated by C*-algebras that are stably isomorphic to closed two-sided ideals of the given algebra. We then use this result to prove that several classes of C*-algebras are closed under crossed products by finite group actions with this Rokhlin property.

scholarly journals Countable sections for locally compact group actions

1992 ◽  
Vol 12 (2) ◽  
pp. 283-295 ◽  
Author(s):  
Alexander S. Kechris
Keyword(s):  
Compact Group ◽  
Equivalence Relation ◽  
Group Actions ◽  
Locally Compact Group ◽  
Borel Space ◽  
Locally Compact ◽  
Borel Equivalence Relation ◽  
Measurable Section ◽  
Singular Action

AbstractIt has been shown by J. Feldman, P. Hahn and C. C. Moore that every non-singular action of a second countable locally compact group has a countable (in fact so-called lacunary) complete measurable section. This is extended here to the purely Borel theoretic category, consisting of a Borel action of such a group on an analytic Borel space (without any measure). Characterizations of when an arbitrary Borel equivalence relation admits a countable complete Borel section are also established.

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