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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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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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*-...
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Relative de Rham Cohomology groups of k-algebra

Let $A$ be a commutative unital $k$-algebra. Then we have de Rham complex given as: $C_{\ast}(A)$ : $ 0 \rightarrow A \rightarrow \Omega_{A \lvert k}^{1} \rightarrow \Omega_{A \lvert k}^{2} \...
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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 ...
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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 $L^...
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$*$-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 $...
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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 ...
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KK-theoretical proof of Atiyah-Singer index theorem

Does anyone know of any detailed proof of the Atiyah-Singer Index Theorem using KK-theory/ Kasparov products? References to any papers and textbooks are greatly appreciated. Thanks!
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Functional calculus for the Dolbeault operator over Hilbert C*-modules

$\newcommand{\odd}{\mathrm{odd}}\newcommand{\even}{\mathrm{even}}$Let $X$ be a complex manifold, you can assume it's compact, if necessary. We have the Dolbeault complex $$0 \rightarrow \mathcal{A}^{0,...
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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 ...
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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 ...
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Bass and Quillen K-theory

What is the difference of Bass and Quillen K-theory groups of a ring $R$. More concretely, what is $K^{Bass}_{i}(R)$? does it equal to $K^{Quillen}_{i}(R)$ if $R$ is a regular ring ? Does it make ...
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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 ...
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Différences between KKO and KKR in Kasparov theory

In Kasparov article : The operator K functor and extensions of $C^*$algebras there is the definition of the two bifunctors $KKO : ralg^{op} \times ralg \to Ab$ and $KKR : Ralg^{op}_r \times Ralg_r \to ...
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3 votes
1 answer
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Self adjoint operators in Kasparov-Modules

In Blackadars book in 17.4.2 it says that for each element $x \in KK(A,B)$ there is a Kasparov module $(E,\pi ,T)$ such that $T=T^*$. Now, the argument for that is that if $(E,\pi, T)$ is any Kasparov-...
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Separable $\sigma$-unital sub-$C^*$-Algebras

Let $A$ be a $\mathbb{Z}_2$-graded $C^*$-Algebra. Then we can take the direct limit $$ colim_{A_\sigma} KK_*(\mathbb{C}, A_\sigma) $$ over all $\sigma$-unital graded $C^*$-sub-algebras $A_\sigma \...
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1 answer
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Homotopy equivalence of Kasparov's $KK$-Theory

The homotopy relation of Kasparov-Cycles is definied in Blackadar's book in 17.2.2. It is an equivalence relation. However, I really don't see a good argument for transitivity and can't find any ...
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Kasparov's descent homomorphism for higher KK groups

I am currently trying to understand equivariant $KK$-theory. I think I roughly get the idea of Kasparov's descent homomorphism $$KK^G(A,B) \rightarrow KK(A \rtimes G,B \rtimes G).$$ but what still ...
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The Green-Julg Theorem

I am currently trying to understand the general Green-Julg theorem, where $G$ is a compact group, $A$ and $B$ are $G$-$C^*$-algebras, and where $G$ acts trivially on $A$. The Green-Julg theorem states ...
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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 ...
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5 votes
1 answer
365 views

reference for KK theory

I wanted to ask you, if you have any good references (book or pdf) to learn about the KK theroy of Kasparov. I think the presentation of Blackadar is too close from the commutative theory. I was ...
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6 votes
2 answers
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Differential structures and K-homology groups

What is an example of a (compact) manifold, which has two non-equivalent differential structures such that the K-homology groups are non-isomorphic? If no such example exists, i.e. "K-homology does ...
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The structure map of topological K-theory

This may be a silly question but I don't know the answer. I know the construction of (equivariant) K-spectrum $KU_G$ and the periodicity of (equivariant) K-theory. But I don't know its structure maps ...
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When is a cycle in $KK^G(A,A)$ with zero operator the identity cycle?

Given a cycle of the form $(\pi,H,0)$ in $KK^G(A,A)$, when is it equivalent to the identity cycle $1_A=(i_A,A,0)$? The operator $T=0$, and $\pi:A \rightarrow L(H)$ may be injective. Any criterions ...
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Is $KK^G(\mathbb{C}^n,B)$ countably additive in $B$ and countable?

Let $G$ be a finite discrete groupoid, $A=\mathbb{C}^n$ a finite dimensional, commutative $C^*$-algebra and assume we have given a $G$-action on $A$. Note that the action of $G$ on $\mathbb{C}^n=C_0(\{...
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4 votes
1 answer
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Definition of homotopy between Kasparov modules

I'm trying to understand the definition of homotopy between Kasparov modules as presented in Blackadar's book on K-theory for operator algebras. $A,B$ will be C*-algebras, while $E$ will denote a ...
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0 answers
281 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 ...
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23 votes
3 answers
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KK-theory as a stable infinity-category and KU Mod

The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it ...
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5 votes
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KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...
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