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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
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