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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Generalized open mapping theorems for bilinear maps, with an application to operator algebras


Author: P. G. Dixon
Journal: Proc. Amer. Math. Soc. 104 (1988), 106-110
MSC: Primary 46A30; Secondary 47A65, 47D25
MathSciNet review: 958052
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Abstract: Cohen [4] gave an example of a surjective bilinear mapping between Banach spaces which was not open, and Horowitz [8] gave a much simpler example. We build on Horowitz' example to produce a similar result for bilinear mappings such that every element of the target space is a linear combination of $ n$ elements of the range. An immediate application is that Bercovici's construction [1] of an operator algebra with property $ ({\mathbb{A}_1})$ but not $ ({\mathbb{A}_1}(r))$ can be extended to achieve property $ ({\mathbb{A}_{1/n}})$ without $ ({\mathbb{A}_{1/n}}(r))$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1988-0958052-0
PII: S 0002-9939(1988)0958052-0
Keywords: Open mapping theorem, bilinear map, dual algebra
Article copyright: © Copyright 1988 American Mathematical Society