11
votes
Accepted
$*$-algebras, completions, and $K$-theory
Any infinite discrete group $\Gamma$ with Kazhdan's property (T) gives an example. Since it is not amenable, the full and reduced C*-algebras (which are both completions of the group algebra) do not ...
- 201
9
votes
Accepted
Role of the UCT problem in classification theory for C*-algebras
Remark: The UCT-sequence is not correct as you stated it, it does not only involve the $K_0$-groups.
Regarding your second quastion: I highly recommend the book "Classification of nuclear $C^*$-...
- 1,188
9
votes
Accepted
K-Theory of $C^{*}(X)$
The group you describe should be the infinite symmetric group $S_{\infty}$. The $K$-theory of its $C^*$-algebra has been determined by Kerov and Vershik in
The K -functor (Grothendieck group) of the ...
- 7,637
7
votes
Accepted
Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
Yes. There is the "singular chains" functor $C_*:\mathcal{S} \to \mathcal{D}(\mathbb{Z})$, where $\mathcal{S}$ denotes your favorite category of spaces, and $\mathcal{D}(\mathbb{Z})$ is the ...
- 10.8k
6
votes
Accepted
Bass and Quillen K-theory
I read that Bass K-theory is sometimes called the non-connective K-theory
Basically, Quillen $K$-theory can be viewed as a spectrum (in the sense of topology), instead of just as a space (this is an ...
- 892
6
votes
Accepted
example of a non-amenable l.c. group such that $C_r^*(G)$ satisfies the UCT
Both $C^\ast(\mathbb F_2)$ and $C^\ast_r(\mathbb F_2)$ satisfy the UCT. This is the special case of the following:
$\mathbf{Theorem}$. If $G$ and $H$ are countable, discrete, amenable groups, then $C^...
- 2,471
6
votes
Is there a categorical version of the splitting principle?
I don't think there can be a reasonable such splitting principle, even in the weakest sense you asked about (i.e. $K_0(\mathsf R) \to K_0(\mathsf R')$ is injective) and in the case when the ground ...
- 40.2k
4
votes
Accepted
Morita equivalence of the invariant uniform Roe algebra and the reduced group C*-algebra
To define the Roe algebra abstractly, we need a suitable covariant representation of the $\Gamma$-$C^\ast$-algebra $(C_0(X), \Gamma)$ on some Hilbert space $H$.
In concrete situations, one usually ...
- 186
3
votes
Is there a $ H_* H^* $-theory which is naturally a common generalization both of singular homology and de Rham (or singular) cohomology?
Let me give some references for bivariant "de Rham theory" for locally convex and $C^*$-algebras. We know that, roughly speaking, Hochschild homology (equipped with an $S^1$-action) is a &...
- 2,856
3
votes
Comments and reference-request on books for KK-theory
Here is a very rough outline of the proof of the index theorem using KK-theory:
Define $KK_G(A, B)$, where $G$ is a Lie group and $A$ and $B$ are
[adjectives] C*-algebras, and the Kasparov product ...
- 29.2k
3
votes
KK-theoretical proof of Atiyah-Singer index theorem
I find the proof in the recent paper of Kasparov
G. Kasparov. Elliptic and transversally elliptic index theory from the
viewpoint of KK-theory. J. Noncommut. Geom., 10(4):1303–1378, 2016
...
- 658
2
votes
Accepted
description of a map in KK-theory
I don't think that there is anything special about $\mathcal O_\infty$ being used here. One observation that might help is that the map $KK(A, \mathcal O_\infty)\to KK(\mathbb C, \mathcal O_\infty)$ ...
- 830
2
votes
How to define an equivariant Kasparov's KK-theory map?
Isn't this treated in Claude Schochet's 1992 paper "On Equivariant Kasparov Theory and Spanier-Whitehead Duality"? Also, if you want more references, I'd recommend using Google Scholar to ...
- 30.3k
1
vote
Producing $K$-homology cycles from $KK$-cycles
Here are just some trivial observations that came to mind after thinking about this a little longer: You are essentially asking for a canonical class in the $K$-homology group $K^0(B) = KK(B,\mathbb{C}...
- 7,637
1
vote
Différences between KKO and KKR in Kasparov theory
I think the difference is because one has real Clifford algebras $Cl_{p,q}(\mathbb{R})$ (indexed by two variables $p,q$), but only complex Clifford algebras $Cl_n(\mathbb{C})$.
My main concern is ...
- 685
1
vote
Self adjoint operators in Kasparov-Modules
What is meant is that $T$ and $(T+T^*)/2$ differ by a compact operator. More precisely:
$$\pi(a) (T - (T+T^*)/2)= \pi(a) (T/2 - T^*/2) \equiv \pi(a) (T/2-T/2)=0$$
for all $a \in A$ modulo $K(E)$, ...
- 685
1
vote
Homotopy equivalence of Kasparov's $KK$-Theory
It is really the same thing as for ordinary homotopy. Mainly a concatenation.
I will write for a space $X$, $XB$ the algebra of continous functions from $X$ to $B$. All the other notations follow ...
- 659
1
vote
The Green-Julg Theorem
For $G$ a finite group, this is part 3 of Theorem 1.22 in the preprint A stable ∞-category for equivariant KK-theory by Ulrich Bunke, Alexander Engel and Markus Land.
More precisely, they prove an ...
- 556
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