# zbMATH — the first resource for mathematics

Going-down functors, the Künneth formula, and the Baum-Connes conjecture. (English) Zbl 1063.46056
The paper begins with an extension of the result of C. Schochet [Pac. J. Math. 98, 443–458 (1982; Zbl 0439.46043)] on the Künneth formula for $$C^*$$-algebras. Let $$\mathcal N$$ be the class of all $$C^*$$-algebras $$A$$ that satisfy the $$K$$-theory Künneth formula with any $$C^*$$-algebra $$B$$. For a locally compact group $$G$$, an equivariant analogue $${\mathcal N}_G$$ of $$\mathcal N$$ is defined as the class of all $$G$$-algebras $$A$$ that satisfy the (mixed) Künneth formula
$$0\to K_*^{\text{top}}(G;A)\otimes K_*(B)\to K_*^{\text{top}}(G;A\otimes B)\to {\text{Tor}}(K_*^{\text{top}}(G;A),K_*(B))\to 0$$
for any $$C^*$$-algebra $$B$$ with trivial $$G$$-action, where $$\otimes$$ denotes the minimal tensor product. It is shown that $$A\in{\mathcal N}_G$$ iff $$A\rtimes_r G\in{\mathcal N}$$ and that if $$A$$ satisfies
$$(*)$$    $$A\rtimes K\in{\mathcal N}$$ for all compact subgroups $$K\subset G,$$
then $$A\in{\mathcal N}_G$$. In particular, $${\mathcal N}_G$$ contains all type I $$G$$-algebras.
Then the connection between the Künneth formula and the Baum-Connes conjecture for $$G$$ with coefficients is studied. Let $$A$$ satisfy $$(*)$$. Suppose further that $$G$$ satisfies the Baum-Connes conjecture with coefficients in $$A\otimes B$$ for every $$C^*$$-algebra $$B$$ with trivial $$G$$-action. Then $$A\rtimes_r G\in{\mathcal N}$$. As a corollary, it is shown that if $$G$$ satisfies the Baum-Connes conjecture with trivial coefficients, then it satisfies this conjecture with any coefficient $$C^*$$-algebra $$B\in{\mathcal N}$$ with trivial $$G$$-action. Using this corollary, the authors prove the following permanence result: if the groups $$G_1$$ and $$G_2$$ satisfy the Baum-Connes conjecture with trivial coefficients and if $$C^*_r(G_1)$$ or $$C^*_r(G_2)$$ belong to $$\mathcal N$$, then $$G_1\times G_2$$ satisfies the same conjecture. Some other applications are presented.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K99 $$K$$-theory and operator algebras
##### Keywords:
Künneth formula; $$K$$-theory; Baum-Connes conjecture
Zbl 0439.46043
Full Text: