Cyclic homology, tight crossed products, and small stabilizations
Guillermo Cortiñas
Universidad de Buenos Aires, Argentina
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Abstract
In [1] we associated an algebra to every bornological algebra and an ideal to every symmetric ideal . We showed that has -theoretical properties which are similar to those of the usual stabilization with respect to the ideal of the algebra of bounded operators in Hilbert space which corresponds to under Calkin's correspondence. In the current article we compute the relative cyclic homology . Using these calculations, and the results of loc. cit., we prove that if is a -algebra and the symmetric ideal of sequences vanishing at infinity, then is homotopy invariant, and that if , it contains as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem [20] that says that for the ideal of compact operators and the -algebra tensor product , we have . Similarly, we prove that if is a unital Banach algebra and , then is invariant under Hölder continuous homotopies, and that for it contains as a direct summand. These -theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups in terms of for general and . For and general , we further compute the latter groups in terms of algebraic differential forms. We prove that the map is an isomorphism in many cases.
Cite this article
Guillermo Cortiñas, Cyclic homology, tight crossed products, and small stabilizations. J. Noncommut. Geom. 8 (2014), no. 4, pp. 1191–1223
DOI 10.4171/JNCG/184