Cyclic homology, tight crossed products, and small stabilizations

In \verb|arXiv:1212.5901| we associated an algebra $\Gami(\fA)$ to every bornological algebra $\fA$ and an ideal $I_{S(\fA)}\triqui\Gami(\fA)$ to every symmetric ideal $S\triqui\elli$. We showed that $I_{S(\fA)}$ has $K$-theoretical properties which are similar to those of the usual stabilization with respect to the ideal $J_S\triqui\cB$ 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 relative cyclic homology $HC_*(\Gami(\fA):I_{S(\fA)})$. Using these calculations, and the results of \emph{loc. cit.}, we prove that if $\fA$ is a $C^*$-algebra and $c_0$ the symmetric ideal of sequences vanishing at infinity, then $K_*(I_{c_0(\fA)})$ is homotopy invariant, and that if $*\ge 0$, it contains $K^{\top}_*(\fA)$ 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 $\fA\sotimes\cK$, we have $K_*(\fA\sotimes\cK)=K^{\top}_*(\fA)$. Similarly, we prove that if $\fA$ is a unital Banach algebra and $\ell^{\infty-}=\bigcup_{q<\infty}\ell^q$, then $K_*(I_{\ell^{\infty-}(\fA)})$ is invariant under H\"older continuous homotopies, and that for $*\ge 0$ it contains $K^{\top}_*(\fA)$ as a direct summand. These $K$-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups $HC_*(\Gami(\fA):I_{S(\fA)})$ in terms of $HC_*(\elli(\fA):S(\fA))$ for general $\fA$ and $S$. For $\fA=\C$ and general $S$, we further compute the latter groups in terms of algebraic differential forms. We prove that the map $HC_n(\Gami(\C):I_{S(\C)})\to HC_n(\cB:J_S)$ is an isomorphism in many cases.