# Questions tagged [kk-theory]

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28
questions

**4**

votes

**0**answers

64 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*-...

**2**

votes

**0**answers

61 views

### 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} \...

**4**

votes

**1**answer

191 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 ...

**8**

votes

**1**answer

246 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

168 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 $...

**4**

votes

**1**answer

176 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 ...

**7**

votes

**1**answer

359 views

### 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!

**5**

votes

**0**answers

112 views

### Functional Calculus for the Dolbeault Operator over Hilbert C*-modules

Let $X$ be a complex manifold, you can assume it's compact, if necessary. We have the Dolbeault complex $$0 \rightarrow \mathcal{A}^{0,0} \xrightarrow{\bar{\partial}} \mathcal{A}^{0,1} \xrightarrow{\...

**8**

votes

**1**answer

442 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 ...

**1**

vote

**1**answer

180 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 ...

**4**

votes

**1**answer

340 views

### 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 ...

**4**

votes

**0**answers

191 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 ...

**2**

votes

**1**answer

141 views

### 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 ...

**3**

votes

**1**answer

138 views

### 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-...

**2**

votes

**0**answers

170 views

### 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 \...

**2**

votes

**1**answer

225 views

### 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 ...

**1**

vote

**0**answers

95 views

### 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 ...

**2**

votes

**0**answers

194 views

### 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 ...

**7**

votes

**1**answer

297 views

### 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 ...

**4**

votes

**1**answer

301 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 ...

**6**

votes

**2**answers

232 views

### 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 ...

**5**

votes

**1**answer

227 views

### 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 ...

**0**

votes

**0**answers

54 views

### 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 ...

**0**

votes

**0**answers

70 views

### 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(\{...

**3**

votes

**1**answer

199 views

### 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 ...

**0**

votes

**0**answers

278 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 ...

**23**

votes

**3**answers

913 views

### 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 ...

**5**

votes

**0**answers

344 views

### 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 ...