scholarly journals Equivariant KK-theory for generalised actions and Thom isomorphism in groupoid twisted K-theory

2013 ◽  
Vol 13 (1) ◽  
pp. 83-113 ◽  
Author(s):  
El-Kaïoum M. Moutuou
Keyword(s):  
Vector Bundles ◽  
Isomorphism Theorem ◽  
Locally Compact ◽  
Bott Periodicity ◽  
C Algebras ◽  
Thom Isomorphism ◽  
K Theory ◽  
Thom Isomorphism Theorem

AbstractWe develop equivariant KK–theory for locally compact groupoid actions by Morita equivalences on real and complex graded C*-algebras. Functoriality with respect to generalised morphisms and Bott periodicity are discussed. We introduce Stiefel-Whitney classes for real or complex equivariant vector bundles over locally compact groupoids to establish the Thom isomorphism theorem in twisted groupoid K–theory.

scholarly journals The equivariant Thom isomorphism theorem

1992 ◽  
Vol 152 (1) ◽  
pp. 21-39
Author(s):  
Steven Costenoble ◽  
Stefan Waner
Keyword(s):  
Isomorphism Theorem ◽  
Thom Isomorphism ◽  
Thom Isomorphism Theorem

scholarly journals Groupoid algebras as Cuntz-Pimsner algebras

2017 ◽  
Vol 120 (1) ◽  
pp. 115 ◽  
Author(s):  
Adam Rennie ◽  
David Robertson ◽  
Aidan Sims
Keyword(s):  
Exact Sequence ◽  
Locally Compact ◽  
C Algebra ◽  
C Algebras ◽  
K Theory ◽  
Étale Groupoid

We show that if $G$ is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle $c\colon G\to\mathbb{Z}$, then the reduced $C^*$-algebra of $G$ can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced $C^*$-algebra of the kernel $G_0$ of $c$. If the full and reduced $C^*$-algebras of $G_0$ coincide, we deduce that the full and reduced $C^*$-algebras of $G$ coincide. We obtain a six-term exact sequence describing the $K$-theory of $C^*_r(G)$ in terms of that of $C^*_r(G_0)$.

scholarly journals On the Classification of Simple Stably Projectionless C*-Algebras

2002 ◽  
Vol 54 (1) ◽  
pp. 138-224 ◽  
Author(s):  
Shaloub Razak
Keyword(s):  
Building Blocks ◽  
Isomorphism Theorem ◽  
Inductive Limits ◽  
C Algebras ◽  
Distinguished Subset ◽  
K Theory

AbstractIt is shown that simple stably projectionless C*-algebras which are inductive limits of certain specified building blocks with trivial K-theory are classified by their cone of positive traces with distinguished subset. This is the first example of an isomorphism theorem verifying the conjecture of Elliott for a subclass of the stably projectionless algebras.

On the Thom Isomorphism Theorem

1962 ◽  
Vol 58 (2) ◽  
pp. 206-208 ◽  
Author(s):  
W. H. Cockcroft
Keyword(s):  
Commutative Ring ◽  
Unit Element ◽  
Isomorphism Theorem ◽  
Thom Isomorphism ◽  
Thom Isomorphism Theorem

In what follows (E, p, B) will always denote a fibering in the sense of Serre, † with arc-wise connected base B, projection p:E → B, and fibre F = p−1(b), b ɛ B. Coefficients will always be taken in a commutative ring A with a unit element, and the cohomology will be singular (cubical), as in (3).

An Introduction to K-Theory for C*-Algebras

2000 ◽  
Author(s):  
M. Rørdam ◽  
F. Larsen ◽  
N. Laustsen
Keyword(s):  
C Algebras ◽  
K Theory

scholarly journals Graded K-theory and K-homology of relative Cuntz–Pimsner algebras and graph C⁎-algebras

2021 ◽  
Vol 496 (2) ◽  
pp. 124822
Author(s):  
Quinn Patterson ◽  
Adam Sierakowski ◽  
Aidan Sims ◽  
Jonathan Taylor
Keyword(s):  
C Algebras ◽  
K Theory

scholarly journals K-theory for certain group C*-algebras

1983 ◽  
Vol 151 (0) ◽  
pp. 209-230 ◽  
Author(s):  
E. Christopher Lance
Keyword(s):  
C Algebras ◽  
K Theory

scholarly journals ON THE CLASSIFICATION OF NUCLEAR C*-ALGEBRAS

2002 ◽  
Vol 85 (1) ◽  
pp. 168-210 ◽  
Author(s):  
MARIUS DADARLAT ◽  
SØREN EILERS
Keyword(s):  
Subject Classification ◽  
Unified Approach ◽  
C Algebras ◽  
Starting Point ◽  
K Theory

We employ results from KK-theory, along with quasidiagonality techniques, to obtain wide-ranging classification results for nuclear C*-algebras. Using a new realization of the Cuntz picture of the Kasparov groups we show that two morphisms inducing equal KK-elements are approximately stably unitarily equivalent. Using K-theory with coefficients to associate a partial KK-element to an approximate morphism, our result is generalized to cover such maps. Conversely, we study the problem of lifting a (positive) partial KK-element to an approximate morphism. These results are employed to obtain classification results for certain classes of quasidiagonal C*-algebras introduced by H. Lin, and to reprove the classification of purely infinite simple nuclear C*-algebras of Kirchberg and Phillips. It is our hope that this work can be the starting point of a unified approach to the classification of nuclear C*-algebras.2000 Mathematical Subject Classification: primary 46L35; secondary 19K14, 19K35, 46L80.

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