Journal of Operator Theory

Volume 59, Issue 1, Winter 2008 pp. 53-68.

Contractive perturbations in $C^*$-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, United Kingdom

Summary: We characterize various objects in a $C^*$-algebra $\mathcal{A}$ in terms of the size and the location of the contractive perturbations. We prove that if $\mathcal{S}$ is a precompact subset of the unit ball of $\mathcal{A}$, there exists a faithful representation $\pi$ of $\mathcal{A}$ such that $\pi(a)$ is compact for each $a\in \mathcal{S}$ if and only if $\mathrm{cp}^2(\lambda\mathcal{S})$ is compact, for each $0<\lambda<1$. We provide a geometric characterization of the hereditary $C^*$-subalgebras and the essential ideals of $\mathcal{A}$, as well as of any separable $C^*$-algebra within its multiplier algebra. We present examples showing that the notion of contractive perturbations is not appropriate for the description of compact operators on a general Banach space.

Keywords: $C^*$-algebra, compact operator, contractive perturbation, hereditary $C^*$- subalgebra, essential ideal, face.

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