Questions tagged [kk-theory]
KK-theory is a common generalization both of K-homology and K-theory as an additive bivariant functor on separable C*-algebras.
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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-...
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Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
It is known that $K_* K^* $-theory is a common generalization both of $K$-homology and $K$-theory as an additive bivariant functor on separable C*-algebras.
Is it possible to construct a $ H_* H^* $-...
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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 \...
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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 ...
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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 $KK(A, C(S))$, ...
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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 ...
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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 ...
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Comments and reference-request on books for KK-theory
I am looking for a good reference to learn Kasparov's KK-theory, where my motivation is to understand (and hopefully can do something about) the Atiyah-Singer index theorem in terms of KK-theory.
I ...
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About the algebraic structure of the $G$-equivariant $KK$-theory
Let $ G $ be a second countable locally compact group.
Let $ A $ and $ B $ be two $G$-$C^*$-algebras.
Let $ KK^G (A, B) $ be the $G$-equivariant $KK$-theory of the pair $ (A, B) $.
Could you tell me ...
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Is there a categorical version of the splitting principle?
One of many places we see a "splitting principle" at work is in the category $\mathsf{Vect}(X)$ of complex vector bundles over a compact connected Hausdorff space $X$. For any object $E$ ...
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Solution without using any k-theory tools
Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...
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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^{...
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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 ...
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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 ...
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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} ( ...
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