All Questions
Tagged with c-star-algebras kk-theory
20 questions
0
votes
0
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44
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How to define a family of Hilbert $A-B$-bimodules $ \pi \ : \ M \to X $, parametrized by a $C^*$-algebra $X$?
Let $A$ and $B$ two $ C^* $ - algebras.
I would like to define a functor $ X \to \mathrm{Bimod}_{A,B} (X) $ which associate to any object $X$, the set of isomorphism classes of a family of Hilbert $A-...
- 595
1
vote
0
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83
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How to define explicitly the Kasparov product $ x \otimes_B y \in KK_{i+j}^G (A,C) $ of $x \in KK_i^G (A,B)$ and, $y \in KK_j^G (B,C)$?
Let $A,B,C$ be separable $G-C^*$ - algebras. Then there is a biadditive pairing for $i,j \in \mathbb{Z}_2$,
$$ KK_i^G (A,B) \times KK_j^G (B,C) \to KK_{i+j}^G (A,C) $$
If $x \in KK_i^G (A,B)$ and, $y \...
- 595
7
votes
0
answers
160
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Maps in the Künneth theorem for K-theory of C*-algebras
The following is named the Künneth theorem for tensor products in the book by Blackadar on K-theory for operator algebras: If $A$ and $B$ are C*-algebras and $A$ is in the bootstrap class, then there ...
- 2,998
1
vote
0
answers
110
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Formula for the KK-theory groups $KK(A, C(S))$
I am studying $C^*$-algebras and their KK-theory. Let $A$ be a (unital if you wish) $C^*$-algebra and $S$ be a compact Hausdorff space. I am interested in computing the KK-theory groups $KK(A, C(S))$, ...
- 1,485
11
votes
0
answers
379
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Why are projectionless $C^*$-algebras important (Kadison's conjecture)
It was considered an important result for a long time to show that the reduced $C^*$-algebra of the free group $C^*_r(F_2)$ has no nontrivial projections. I believe this is also known as Kadison's ...
- 211
5
votes
0
answers
137
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C^*-algebra theory with all the Koszul signs
I was wondering if someone knows of a reference in which $\mathbb{Z}_2$-graded $C^*$-algebra theory is developed using the sign convention $(ab)^* = (-1)^{|a||b|}b^* a^*$. I would be most enthusiastic ...
- 357
2
votes
0
answers
124
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Representation of $C^{*} (S_{\infty})$
I was wondering what is the group $C^{*}$-algebra of infinite symmetric group?
Mainly, I was trying to calculate the k-theory of $C^{*}$-algebra of infinite symmetric group and I found K-Theory of $C^{...
- 581
2
votes
1
answer
352
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K-Theory of $C^{*}(X)$
I'm new to K-Theory for $C^{*}$-algebra and $C^{*}$-algebra of groups.
If $X$ is the group of finite support bijections of natural numbers then what is the K-Theory of $C^{*}(X)$?
I was planning to ...
- 581
0
votes
1
answer
109
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How to define an equivariant Kasparov's KK-theory map?
I'm looking for some references about how to construct an equivariant Kasparov's KK-theory map $$ \psi \ : \ KK^{G_{1}} ( A,B ) \to KK^{G_{2}} ( C,D ) $$, where, $ G_1 $ and $ G_2 $ are two distinct ...
- 325
1
vote
0
answers
69
views
A map from a $ G_1 $ - equivariant KK-theory of Kasparov, to a $ G_2 $ - equivariant KK-theory of Kasparov
Let $ G $ be a locally compact group.
Let $ H $ and $ K $ be two normal subgroups of $ G $.
In order to construct a map, $$ \psi \ : \ \ F(G/H,G/K) \to F(G/K,G/H) $$
where, $$ F(G/H,G/K) = KK^{G/H} ( ...
- 325
3
votes
0
answers
129
views
Another way for defining $K_1$ group for a C*-algebra
Thank you for answering my question.
I have another question about the $K_1$ group. As you may know, some books define the $K_1$ group like below:
Also, it defines the $K_0$ group for an arbitrary C*-...
- 581
4
votes
1
answer
277
views
Producing $K$-homology cycles from $KK$-cycles
For two unital (separable) $C^*$-algebras $A$ and $B$, let $(H,\rho,F)$ be a $KK$-cycle in the sense of Kasparov, or in the sense of Wikipedia :)
I wonder if there us a natural way to "forget" the ...
- 993
8
votes
1
answer
356
views
Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra
In his paper "Comparing analytic assemby maps", J. Roe considers a proper and cocompact action of a countable group $\Gamma$ on a metric space $X$. He constructs the Hilbert $C^*_r(\Gamma)$-module $L^...
7
votes
1
answer
219
views
$*$-algebras, completions, and $K$-theory
What is an example of a $*$-algebra $\cal{A}$, which admits two non-equivalent norms $\| \cdot \|_1$ and $\| \cdot \|_2$, with respect to which we can complete $\cal{A}$ to give two $C^*$-algebras $...
- 993
5
votes
1
answer
290
views
example of a non-amenable l.c. group such that $C_r^*(G)$ satisfies the UCT
Are there known any examples of non-amenable locally compact (or more restrictive, non-amenable discrete) groups $G$ for which the reduced group $C^*$-algebra $C_r^*(G)$ satisfies the universal ...
user62639
8
votes
1
answer
725
views
Role of the UCT problem in classification theory for C*-algebras
Elliott's program for nuclear C*-algebras deals with the problem of classifying nuclear C*-algebras by K-theoretical invariants. A major open question in this context is the UCT problem.
A separable ...
- 741
1
vote
1
answer
269
views
description of a map in KK-theory
The following situation is given: Let $A$ be a unital, separable, nuclear $C^*$-Algebra, $i:\mathbb{C}\to A$ the unital embedding. All $C^*$-algebras are considered as trivially graded. Consider the ...
user62639
4
votes
0
answers
389
views
Künneth formula for $C^*$ algebras, equivalent condition for full generality
I'm crrently reading the paper about the Künneth-theorem for $C^*$-algebras: http://msp.org/pjm/1982/98-2/pjm-v98-n2-p15-s.pdf and I'm trying to understand remark 4.9. I henceforth asumme that $A$ is ...
- 81
7
votes
1
answer
454
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Relation between the UCT and Künneth ($C^*$-algebras)
I have a (maybe dumb) question about the relation between the Künneth theorem and the Theorem-universal coefficient theorem (UCT for short) in $KK$-theory (for the setting see "The Künneth theorem and ...
user62639
0
votes
0
answers
293
views
Lifting triangles in K-theory to KL-groups
Let $X$ and $Y$ be finite simplicial complexes (or $CW$-complexes) so that $Y\subseteq X$. Let $s\colon C(X)\to C(Y)$ be the map given by restriction. In particular $K_{*}(C(X))$ and $K_{*}(C(Y))$ are ...
