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c*-bundle

Proceedings of the Edinburgh Mathematical Society (1)

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On derivations and elementary operators on C*-algebras
Part of
Ilja Gogić
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 56 / Issue 2 / June 2013

Published online by Cambridge University Press:

21 March 2013, pp. 515-534
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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.