Operator ideals and assembly maps in K-theory.

GUILLERMO CORTIÑAS; GISELA TARTAGLIA

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2013 vol. 142 p. 1089 - 1099

0002-9939

Let $cB$ be the ring of bounded operators in a complex, separable Hilbert space. For $p>0$ consider the Schatten ideal $cL^p$ consisting of those operators whose sequence of singular values is $p$-summable; put $cS=igcup_pcL^p$. Let $G$ be a group and $cyc$ the family of virtually cyclic subgroups. Guoliang Yu proved that the $K$-theory assembly map [ H_*^G(cE(G,cyc),K(cS)) o K_*(cS[G]) ] is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coefficients $cS$ and the use of some results about algebraic $K$-theory of operator ideals and about controlled topology and coarse geometry. In this paper we give a different proof of Yu´s result. Our proof uses the usual Chern character to cyclic homology. Like Yu´s, it relies on results on algebraic $K$-theory of operator ideals, but no controlled topology or coarse geometry techniques are used. We formulate the result in terms of homotopy $K$-theory. We prove that the rational assembly map [ H_*^G(cE(G,in),KH(cL^p))otimesQ o KH_*(cL^p[G])otimesQ ] is injective. We show that the latter map is equivalent to the assembly map considered by Yu, and thus obtain his result as a corollary.