Cyclic homology, tight crossed products, and small stabilizations

Abstract

In [1] we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C 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. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ 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.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 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 differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases. Mathematics In [1] (arXiv:1212.5901) we associated an algebra 1.A/ to every bornological algebra A and an ideal IS.A/ C 1.A/ to every symmetric ideal S C `1. We showed that IS.A/ has K-theoretical properties which are similar to those of the usual stabilization with respect to the ideal JS C 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. 1.A/ W IS.A//. Using these calculations, and the results of loc. cit., we prove that if A is a C -algebra and c0 the symmetric ideal of sequences vanishing at infinity, then K.Ic0.A// is homotopy invariant, and that if 0, it contains K top .A/ as a direct summand. This is a weak analogue of the Suslin–Wodzicki theorem ([20]) that says that for the ideal K D Jc0 of compact operators and the C -algebra tensor product A ˝ K, we have K.A ˝ 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.I`1.A// is invariant under Hölder continuous homotopies, and that for 0 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 differential forms. We prove that the map HCn. 1.C/ W IS.C// ! HCn.B W JS / is an isomorphism in many cases.