Cyclic homology, tight crossed products, and small stabilizations

In [ 1] we associated an algebra Gamma(infinity)(U) to every bornological algebra U and an ideal I-S(U) del Gamma(infinity) to every symmetric ideal S del l(infinity). We showed that I-S ((U)) has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal J(S) del B of the algebra B of bounded operators in Hilbert space which corresponds to S under Calkin's correspondence. In the current article we compute the relative cyclic homology HC*(Gamma(infinity)):I-s(U)) Using these calculations, and the results of loc. cit., we prove that if A is a C* - algebra and c(0) the symmetric ideal of sequences vanishing at infinity, then K*(I-c0(U)) is homotopy invariant, and that if * >= 0, it contains K-*(top)(U) as a direct summand. This is a weak analogue of the Suslin- Wodzicki theorem ([ 20]) that says that for the ideal K = J(c0) of compact operators and the C * - algebra tensor product AK, we have KA K / D K top A Similarly, we prove that if A is a unital Banach algebra and ` 1 D S q< 1 q, then K I1.A // is invariant under H lder continuous homotopies, and that for0 it contains K-top A / as a direct summand. These K- theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC . _ 1. A / W IS. A // in terms of HC 1. A /W S. A // for general A and S. For A D C and general S, we further compute the latter groups in terms of algebraic di ff erential forms. We prove that the map HCn.1. C/W IS. C // HCn. B W JS/ is an isomorphism in many cases.