Journal of Operator Theory
Volume 59, Issue 1, Winter 2008 pp. 53-68.
Contractive perturbations in C∗-algebrasAuthors: 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 \ca in terms of the size and the location of the contractive perturbations. We prove that if \cs is a precompact subset of the unit ball of \ca, there exists a faithful representation π of \ca such that π(a) is compact for each a∈\cs if and only if \cp2(λ\cs) is compact, for each 0<λ<1. We provide a geometric characterization of the hereditary C∗-subalgebras and the essential ideals of \ca, 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.
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