Cyclic homology, tight crossed products, and small stabilizations

Abstract

In [1] we associated an algebra ${\Gamma }^{\infty }(\mathfrak{A})$ to every bornological algebra $\mathfrak{A}$ and an ideal $I_{S(\mathfrak{A})}⊲{\Gamma }^{\infty }(\mathfrak{A})$ to every symmetric ideal $S⊲{\ell }^{infty}$. 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 $J_S⊲\mathcal{B}$ 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 $H{C}_{\ast }({\Gamma }^{\infty }(\mathfrak{A}):I_{S(\mathfrak{A})})$. Using these calculations, and the results of loc. cit., we prove that if $\mathfrak{A}$ is a $C^{\ast }$-algebra and $c_0$ the symmetric ideal of sequences vanishing at infinity, then $K_{\ast }(I_{c_0(\mathfrak{A})})$ is homotopy invariant, and that if $\ast \ge 0$, it contains $K_{\ast }^{\top }(\mathfrak{A})$ 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 $C^{\ast }$-algebra tensor product $\mathfrak{A}\stackrel{\sim }{\otimes }\mathcal{K}$, we have $K_{\ast }(\mathfrak{A}\stackrel{\sim }{\otimes }\mathcal{K})=K_{\ast }^{\top }(\mathfrak{A})$. Similarly, we prove that if $\mathfrak{A}$ is a unital Banach algebra and ${\ell }^{\infty -}={\bigcup }_{q<\infty }{\ell }^q$, then $K_{\ast }(I_{{\ell }^{\infty -}(\mathfrak{A})})$ is invariant under Hölder continuous homotopies, and that for $\ast \ge 0$ it contains $K_{\ast }^{\top }(\mathfrak{A})$ as a direct summand. These $K$-theoretic results are obtained from cyclic homology computations. We also compute the relative cyclic homology groups $H{C}_{\ast }({\Gamma }^{\infty }(\mathfrak{A}):I_{S(\mathfrak{A}}))$ in terms of $H{C}_{\ast }({\ell }^{infty}(\mathfrak{A}):S(\mathfrak{A}))$ 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 $H{C}_n({\Gamma }^{\infty }(\mathbb{C}):I_{S(\mathbb{C})})\to H{C}_n(\mathcal{B}:J_S)$ is an isomorphism in many cases.