Cyclic homology, tight crossed products, and small stabilizations

الأصل · EN

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 ℓ∞⁻=<∞ℓq, then K*(Iℓ∞⁻()) is invariant under Hölder 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.