Homotopy invariance through small stabilizations
type: Item
instance of: scholarly article
title: Homotopy invariance through small stabilizations
zbMATH Open document ID: 1347.55006
author: Beatriz Abadie
author: G. Cortiñas
published in: Journal of Homotopy and Related Structures
publication date: 2015-09-21T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.5901
review text: 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 \(c_0\),…
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 55N15
Mathematics Subject Classification ID: 19D55
Mathematics Subject Classification ID: 18F25
zbMATH DE Number: 6483846
zbMATH Keywords: operator ideal
zbMATH Keywords: Calkin's theorem
zbMATH Keywords: crossed product
zbMATH Keywords: Karoubi's cone
zbMATH Keywords: \(K\)-theory
MaRDI profile type: Publication
OpenAlex ID: W3098858264
arXiv ID: 1212.5901
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DOI: 10.1007/S40062-013-0069-9
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description: scientific article