# Cyclic homology, tight crossed products, and small stabilizations

### Guillermo Cortiñas

Universidad de Buenos Aires, Argentina

## Abstract

In [1] we associated an algebra $Γ_{∞}(A)$ to every bornological algebra $A$ and an ideal $I_{S(A)}⊲Γ_{∞}(A)$ to every symmetric ideal $S⊲ℓ_{infty}$. We showed that $I_{S(A)}$ has $K$-theoretical properties which are similar to those of the usual stabilization with respect to the ideal $J_{S}⊲B$ 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 $HC_{∗}(Γ_{∞}(A):I_{S(A)})$. Using these calculations, and the results of *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 $∗≥0$, it contains $K_{∗}(A)$ as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem [20] that says that for the ideal $K=J_{c_{0}}$ of compact operators and the $C_{∗}$-algebra tensor product $A⊗∼ K$, we have $K_{∗}(A⊗∼ K)=K_{∗}(A)$. Similarly, we prove that if $A$ is a unital Banach algebra and $ℓ_{∞−}=⋃_{q<∞}ℓ_{q}$, then $K_{∗}(I_{ℓ_{∞−}(A)})$ is invariant under Hölder continuous homotopies, and that for $∗≥0$ it contains $K_{∗}(A)$ as a direct summand. These $K$-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups $HC_{∗}(Γ_{∞}(A):I_{S(A}))$ in terms of $HC_{∗}(ℓ_{infty}(A):S(A))$ for general $A$ and $S$. For $A=C$ and general $S$, we further compute the latter groups in terms of algebraic differential forms. We prove that the map $HC_{n}(Γ_{∞}(C):I_{S(C)})→HC_{n}(B:J_{S})$ 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