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Baum-Connes conjecture for some semi-direct products. (English) Zbl 0959.46049
Following a general construction suggested by P. Baum, A. Connes and N. Higson [Contemp. Math. 167, 241-291 (1994; Zbl 0830.46061)] and using the Dirac/dual Dirac method that G. Kasparov designed to prove special cases of the Novikov conjecture the author proves the Baum-Connes conjecture for the reduced $$C^\ast$$-algebra of a second countable locally compact group $$G$$ in the following case. The group $$G$$ is assumed to have a $$\gamma$$-element to make the Kasparov method work. Moreover, $$G$$ is a semi-direct product of a group $$H$$, for which the Baum-Connes conjecture holds, and an amenable group N. Now the conjecture for G with coefficients in a $$G$$-algebra $$B$$ can be reduced to the conjecture for $$H$$ with coefficients in the crossed product of $$B$$ by $$N$$ provided that $$H$$ is almost connected or has a compact-open subgroup.

##### MSC:
 46L80 $$K$$-theory and operator algebras (including cyclic theory) 19K33 Ext and $$K$$-homology 19K35 Kasparov theory ($$KK$$-theory)
Zbl 0830.46061
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##### References:
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