Cyclic homology, tight crossed products, and small stablizations.

GUILLERMO CORTIÑAS

JOURNAL OF NONCOMMUTATIVE GEOMETRY

EUROPEAN MATHEMATICAL SOC

Año: 2014 p. 1191 - 1223

1661-6952

In erb|arXiv:1212.5901| we associated an algebra $Gami(A)$ to every bornological algebra $A$ and an ideal $I_{S(A)} riquiGami(A)$ to everysymmetric ideal $S riquielli$. We showed that $I_{S(A)}$ has $K$-theoretical properties which are similar to those of the projective tensor product$Ahotimes J_S$ with the ideal $J_S riquicB$ of the algebra $cB$ of bounded operators in Hilbert space which corresponds to $S$ under Calkin´s correspondence. In the current article we compute the relativecyclic homology $HC_*(Gami(A):I_{S(A)})$. Using these calculations, and the results of emph{loc. cit.}, we  prove that if $A$ is a $C^*$-algebra and $c_0$ the symmetric ideal of sequences vanishing at infinity, then $K_*(I_{c_0(A)})$ is homotopy invariant and that if $*ge 0$, it contains $K^{ op}_*(A)$ as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem (cite{sw1}) that says that for the ideal $cK=J_{c_0}$ of compact operators and the $C^*$-algebra tensor product $AsotimescK$, we have $K_*(AsotimescK)=K^{ op}_*(A)$. Similarly, we prove that if $A$ is a unital Banach algebraand $ell^{infty-}=igcup_{q<infty}ell^q$, then $K_*(I_{ell^{infty-}(A)})$ is invariant under H"older continuous homotopies and that for $*ge 0$ it contains $K^{ op}_*(A)$as a direct summand. These $K$-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups $HC_*(Gami(A):I_{S(A)})$ in terms of $HC_*(elli(A):S(A))$ for general $A$ and $S$. For $A=C$ and general $S$, we further compute the lattergroups in terms of algebraic differential forms. We prove that the map $HC_n(Gami(C):I_{S(C)}) o HC_n(cB:J_S)$ is an isomorphism in many cases.