Bidual as a weak nonstandard hull

Details Bidual as a weak nonstandard hull Bidual as a weak nonstandard hull

2008-10-17

arxiv.org/abs/0810.3090v1

Mathematics

Functional Analysis

14 pages

Scientific paper

We construct the weak nonstandard hull of a normed linear space X from *X (the nonstandard extension of X) using the weak topology on X. The bidual (i.e. the second dual) X" is shown to be isometrically isomorphic to the weak nonstandard hull of X. Examples and applications to C*-algebras are given, including a simple proof of the Sherman-Takeda Theorem. As a consequence, the weak nonstandard hull of a C*-algebra is always a von Neumann algebra. Moreover a natural representation of the Arens product is given.

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