Homotopy invariance through small stabilizations

Abstract

We associate an algebra $\Gami(\fA)$ to each bornological algebra $\fA$. The algebra $\Gami(\fA)$ contains a two-sided ideal $I_{S(\fA)}$ for each symmetric ideal $S\triqui\elli$ of bounded sequences of complex numbers. In the case of $\Gami=\Gami(\C)$, these are all the two-sided ideals, and $I_S\mapsto J_S=\cB I_S\cB$ gives a bijection between the two-sided ideals of $\Gami$ and those of $\cB=\cB(\ell^2)$. We prove that Weibel's $K$-theory groups $KH_*(I_{S(\fA)})$ are homotopy invariant for certain ideals $S$ including $c_0$ and ${\ell }^p$. Moreover, if either $S=c_0$ and $\fA$ is a local $C^{\ast }$-algebra or $S={\ell }^p,{\ell }^{p\pm }$ and $\fA$ is a local Banach algebra, then $KH_*(I_{S(\fA)})$ contains $K_*^{\top}(\fA)$ as a direct summand. Furthermore, we prove that for $S\in \{c_0,{\ell }^p,{\ell }^{p\pm }\}$ the map $K_*(Γ^\infty(\fA):I_{S(\fA)})\to KH_*(I_{S(\fA)})$ fits into a long exact sequence with the relative cyclic homology groups $HC_*(Γ^\infty(\fA):I_{S(\fA)})$. Thus the latter groups measure the failure of the former map to be an isomorphism.

32 pages. The original paper has been split into two parts, of which this is the first part. The second part is now arXiv:1304.3508

References in corpus (1)

• The obstruction to excision in K-theory and in cyclic homology