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This might be well-known or trivial, but I could not figure out how to fill in the details: For an $S$-algebra $K$ denote its associated multiplicative cohomology theory by $h^*_K$. Suppose that I hav …

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 i …

Let $Y$ be the Fermat quintic, i.e. $Y \subset \mathbb{C}P^4$ is defined by $$ \sum_{i=1}^5 z_i^5 = 0 $$ In Section 5.3 of this paper by Volker Braun the author computes the K-groups of a quotient $X …

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 …

This is not necessarily an answer, but it was too long for a comment: Note that for any separable unital $C^*$-algebra $A$ its suspension $SA := C_0(\mathbb{R}) \otimes A$ is quasidiagonal. This can …

Observe that the $KK$-class $\sigma \in KK_1(A/J,J)$, which you mention in your edited paragraph only depends on the extension $$ 0 \to J \to A \to A/J \to 0 $$ and not on $B$. So we have $\delta_1^n …

I think the key idea is that Connes' Thom isomorphism is itself given by a $KK$-equivalence (see for example Blackadar's book "K-theory for Operator Algebras" - Theorem 19.3.6). This means there are …

This is just an addendum to Sebastian Goette's excellent answer: In fact, you can retrieve the integer-valued function that you mention from the $K$-theory class: Let $\iota_x \colon \{pt\} \to X$ be …

This is more or less an addendum to AlexE's answer: Even though the $K$-homology groups themselves do not depend on the smooth structure there are classes in $K$-theory, which can tell apart some of …

This is an awkwardly backwards question, but bear with me here: Suppose I have a graded ring $R$ with unit, which has an invertible element $u$ in degree $2$. The multiplicative formal group law $f(x …

As David Handelman already pointed out, by continuity of $K$-theory we obtain that $K_0(C(X)) \cong \bigoplus_{\mathbb{N}} \mathbb{Z}$ and $K_1(C(X)) = 0$. Since $C(X)$ is commutative, it lies in the …

I am looking for functorial ways of replacing a commutative $C^*$-algebra $C$ by a simple one, say $A$ , such that the $K$-theory remains unchanged, i.e. $K_*(C) \cong K_*(A)$. I am particularly int …

There is a paper by Jonathan Rosenberg with the title "The K-homology class of the Euler characteristic operator is trivial" see here, which proves that the only information contained in the class of …

There is a version of $KK$-theory for Banach algebras, which was developed by Lafforgue. There also is a paper titled Banach KK-theory and the Baum-Connes conjecture, which is probably relevant for th …

I read in a paper by Christopher Douglas that third cohomology twists of $K$-theory may be interpreted as TMF-classes via a map $K(\mathbb{Z},3) \to TMF$, which is related to String orientations. How …