Journal of Operator Theory

Volume 66, Issue 2, Fall 2011 pp. 235-260.

$s$-Numbers of elementary operators on $C^*$-algebras

Authors: M. Anoussis (1), V. Felouzis (2), and I.G. Todorov (3)
Author institution: (1) Department of Mathematics, University of the Aegean, 832 00 Karlovasi, Samos, Greece
(2) Department of Mathematics, University of the Aegean, 832 00 Karlovasi, Samos, Greece
(3) Department of Pure Mathematics, Queen's University Belfast, Belfast BT7 1NN, U.K.

Summary: We study $s$-functions of elementary operators acting on $C^*$-alge\-bras. The main results are the following: If $\tau$ is any tensor norm and $A,B\in\bfb(\cl H)$ are such that the sequences $s(A),s(B)$ of their singular numbers belong to a tensor stable Calkin space $\fri$ then the sequence of approximation numbers of $A\otimes_{\tau} B$ belongs to $\fri$. If $\mathcal{A}$ is a $C^{*}$-algebra, $\mathfrak{i}$ is a tensor stable Calkin space, ${\rm s}$ is an $s$-number function, and $a_i, b_i \in \mathcal{A},$ $ i=1,2,\dots,m$ are such that $s(\pi(a_i)), s(\pi(b_i)) \in \mathfrak{i}$, $i=1,2,\dots,m$ for some faithful representation $\pi$ of $\cl A$ then ${\rm s}\Big(\sum\limits_{i=1}^{m} M_{a_i,b_i}\Big)\in \mathfrak{i}$. The converse implication holds if and only if the ideal of compact elements of $\cl A$ has finite spectrum. We also prove a quantitative version of a result of Ylinen.

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