Journal of Noncommutative Geometry

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Volume 8, Issue 4, 2014, pp. 1191–1223
DOI: 10.4171/JNCG/184

Published online: 2015-02-02

Cyclic homology, tight crossed products, and small stabilizations

Guillermo Cortiñas[1]

(1) Universidad de Buenos Aires, Argentina

In [1] we associated an algebra $\Gamma^\infty (\mathfrak A)$ to every bornological algebra $\mathfrak A$ and an ideal $I_{S(\mathfrak A)}\vartriangleleft \Gamma^\infty (\mathfrak A)$ to every symmetric ideal $S\vartriangleleft \ell^{infty}$. We showed that $I_{S(\mathfrak A)}$ has $K$-theoretical properties which are similar to those of the usual stabilization with respect to the ideal $J_S\vartriangleleft \mathcal B$ of the algebra $\mathcal B$ of bounded operators in Hilbert space which corresponds to $S$ under Calkin's correspondence. In the current article we compute the relative cyclic homology $HC_*(\Gamma^\infty (\mathfrak A):I_{S(\mathfrak A)})$. Using these calculations, and the results of loc. cit., we prove that if $\mathfrak A$ is a $C^*$-algebra and $c_0$ the symmetric ideal of sequences vanishing at infinity, then $K_*(I_{c_0(\mathfrak A)})$ is homotopy invariant, and that if $*\ge 0$, it contains $K^{\top}_*(\mathfrak A)$ as a direct summand. This is a weak analogue of the Suslin-Wodzicki theorem [20] that says that for the ideal $\mathcal K=J_{c_0}$ of compact operators and the $C^*$-algebra tensor product $\mathfrak A\overset{\sim}{\otimes}\mathcal K$, we have $K_*(\mathfrak A\overset{\sim}{\otimes} \mathcal K)=K^{\top}_*(\mathfrak A)$. Similarly, we prove that if $\mathfrak A$ is a unital Banach algebra and $\ell^{\infty-}=\bigcup_{q<\infty}\ell^q$, then $K_*(I_{\ell^{\infty-}(\mathfrak A)})$ is invariant under Hölder continuous homotopies, and that for $*\ge 0$ it contains $K^{\top}_*(\mathfrak A)$ as a direct summand. These $K$-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups $HC_*(\Gamma^\infty (\mathfrak A):I_{S(\mathfrak A}))$ in terms of $HC_*(\ell^{infty}(\mathfrak A):S(\mathfrak A))$ for general $\mathfrak A$ and $S$. For $\mathfrak A=\mathbb C$ and general $S$, we further compute the latter groups in terms of algebraic differential forms. We prove that the map $HC_n(\Gamma^\infty (\mathbb C):I_{S(\mathbb C)})\to HC_n(\mathcal B:J_S)$ is an isomorphism in many cases.

Keywords: Operator ideal, Calkin’s theorem, crossed product, Karoubi’s cone, cyclic homology

Cortiñas Guillermo: Cyclic homology, tight crossed products, and small stabilizations. J. Noncommut. Geom. 8 (2014), 1191-1223. doi: 10.4171/JNCG/184