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Homotopy invariance through small stabilizations

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 $C^*$-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.