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Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

Published online by Cambridge University Press:  11 April 2011

Philip G. Spain
Philip G. Spain
Affiliation:
School of Mathematics and Statistics, University of Glasgow, University Gardens, Glasgow G12 8QW, UK ([email protected])
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Abstract

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Palmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.

When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

Keywords

numerical rangeBoolean algebras of projectionsdual Banach spacesC*-algebras
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 54 , Issue 2 , June 2011 , pp. 515 - 529
Copyright
Copyright © Edinburgh Mathematical Society 2011

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