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On derivations and elementary operators on C*-algebras

Part of: Selfadjoint operator algebras Fiber spaces and bundles Special classes of linear operators

Published online by Cambridge University Press:  21 March 2013

Ilja Gogić
Ilja Gogić*
Affiliation:
Department of Mathematics, University of Zagreb, Bijenička Cesta 30, Zagreb 10000, Croatia (ilja@math.hr)
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Abstract

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Let A be a unital C*-algebra with the canonical (H) C*-bundle over the maximal ideal space of the centre of A, and let E(A) be the set of all elementary operators on A. We consider derivations on A which lie in the completely bounded norm closure of E(A), and show that such derivations are necessarily inner in the case when each fibre of is a prime C*-algebra. We also consider separable C*-algebras A for which is an (F) bundle. For these C*-algebras we show that the following conditions are equivalent: E(A) is closed in the operator norm; A as a Banach module over its centre is topologically finitely generated; fibres of have uniformly finite dimensions, and each restriction bundle of over a set where its fibres are of constant dimension is of finite type as a vector bundle.

Keywords

C*-algebraderivationelementary operatoridealC*-bundlevector bundle

MSC classification

Secondary: 46L05: General theory of $C^*$-algebras 47B47: Commutators, derivations, elementary operators, etc. 46L07: Operator spaces and completely bounded maps 55R05: Fiber spaces 46L08: $C^*$-modules

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 515 - 534
Copyright
Copyright © Edinburgh Mathematical Society 2013

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