Bidual as a weak nonstandard hull

TL;DR

This paper constructs a weak nonstandard hull of a normed space using nonstandard analysis, demonstrating its isometric isomorphism with the bidual, and applies this to C*-algebras to show they form von Neumann algebras.

Contribution

It introduces a novel construction of the weak nonstandard hull and proves its isometric isomorphism with the bidual, with applications to C*-algebras and von Neumann algebras.

Findings

The bidual of a normed space is isometrically isomorphic to its weak nonstandard hull.

The weak nonstandard hull of a C*-algebra is always a von Neumann algebra.

A natural representation of the Arens product is provided.

Abstract

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.