OPERATOR IDEALS AND ASSEMBLY MAPS IN K-THEORY

GUILLERMO CORTIÑAS; GISELA TARTAGLIA

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY

AMER MATHEMATICAL SOC

Lugar: Providence; Año: 2014 vol. 142 p. 1089 - 1099

0002-9939

Let B be the ring of bounded operators in a complex, separable Hilbert space.  For p > 0 consider the Schatten ideal Lp consisting of those  operators whose sequence of singular values is p-summable; put S =cup_p L_p. Let G be a group and Vcyc the family of virtually cyclic subgroups. Guoliang Yu proved that the K-theory assembly map  H^G _*(E(G; Vcyc);K(S)) --> K_*(S[G])  is rationally injective. His proof involves the construction of a certain Chern character tailored to work with coeficients S 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, our proof 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_*(E(G,FIN),KH(L_p)) otimes Q --> KH_*(L_p ^[G]) \otimes Q 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.