Cyclic homology, tight crossed products, and small stabilizations
Abstract
In |arXiv:1212.5901| we associated an algebra () to every bornological algebra and an ideal IS()() to every symmetric ideal S. We showed that IS() has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS of the algebra 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*(():IS()). Using these calculations, and the results of loc. cit., we prove that if is a C*-algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K*(Ic0()) is homotopy invariant, and that if * 0, it contains K*() as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem (sw1) that says that for the ideal =Jc0 of compact operators and the C*-algebra tensor product , we have K*()=K*(). Similarly, we prove that if is a unital Banach algebra and ∞-=q<∞q, then K*(I∞-()) is invariant under H\"older continuous homotopies, and that for * 0 it contains K*() as a direct summand. These K-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups HC*(():IS()) in terms of HC*(():S()) for general and S. For = and general S, we further compute the latter groups in terms of algebraic differential forms. We prove that the map HCn(():IS()) HCn(:JS) is an isomorphism in many cases.
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