Homotopy invariance through small stabilizations
Let \(\mathfrak A\) be a bornological algebra, \(\ell^\infty\) the algebra of bounded sequences, \(\mathcal B = \mathcal B(\ell^\infty)\) the algebra of bounded operators in \(\ell^\infty\), Emb the inverse monoid of all partially defined injections \(f: \mathrm{dom}(f)\subset \mathbb N \to \mathbb N\). To each \(f\in\mathrm{Emb}\) one associates a partial isometry \(U_f\in \mathcal B (\ell^\infty)\) mapping any basis vector \(e_n\) of \(\ell^\infty\) to \(e_{f(n)}\), if \(n\in \mathrm{dom}(f)\) and 0 otherwise. Let each vector \(\alpha\in \ell^\infty\) act on \(\ell^\infty\) itself by the diagonal action \(\mathrm{diag}(\alpha)(e_n) = \alpha(n)e_n\). Let \(\Gamma^\infty\) be the subalgebra of \(\mathcal B\) generated by partial isometries \(U_f, f\in\mathrm{Emb}\) and \(\mathrm{diag}(\alpha), \alpha\in \ell^\infty\). Calkin's correspondence establishes a one-to-one correspondence between two-sided ideals of \(\mathcal B\) and the ideals of \(\ell^\infty\) that are symmetric (i.e. invariant under the action of Emb): a symmetric ideal \(S \triangleleft \ell^\infty\) is mapped to the ideal \(J_S\triangleleft \mathcal B\) of the operators whose sequence of singular values belongs to \(S\). The first result of the paper is Theorem 4.2 stating that the correspondence \(J \mapsto J \cap \Gamma^\infty\) between the set of two-sided ideals \(J\) in \(\mathcal B(\ell^2(\mathbb N))\) and the set of ideals \(I\) in \(\Gamma^\infty\) is a bijection and if the ideal \(S\triangleleft \ell^\infty\) is symmetric then \(I_S= J_S \cap \Gamma^\infty\). The next interesting result is Theorem 8.2 stating that: Weibel's homotopy algebraic K-theory \(\mathfrak A \mapsto KH_*(I_{c_0(\mathfrak A)})\) from the category BAlg of bornological algebras to the category \(\mathfrak{Ab}\) of abelian groups is invariant under continuous homotopy. Let \(S(\mathfrak A) = S \tilde{\otimes} \mathfrak A\). The third interesting result is Theorem 8.5 stating that for \(S\) being one of the ideals \(c_0\), \(\ell^p\), \(\ell^{p+} = \cap_{s>p} \ell^s , \ell^{p-} = \cup_{r <p} \ell^r (0<p<\infty)\) of \(\ell^\infty\), there are long exact sequences between Weibel's homotopy algebraic K-theory \(KH_*\), Quillen's K-theory \(K_*\) and cyclic homology \(HC_*\): \[ KH_{n+1}(I_{S(\mathfrak A)}) \to HC_{n-1}(I_{S(\mathfrak A)}) \to K_n(I_{S(\mathfrak A)}) \to KH_n(I_{S(\mathfrak A)}) \] and \[ KH_{n+1}(I_{S(\mathfrak A)}) \to HC_{n-1}(\Gamma^\infty(\mathfrak A): I_{S(\mathfrak A)}) \to K_n(\Gamma^\infty(\mathfrak A):I_{S(\mathfrak A)}) \to KH_n(I_{S(\mathfrak A)}) \]
scientific article