Nonstandard Hulls of C*-Algebras

TL;DR

This paper investigates the properties of C*-algebras derived from nonstandard hull constructions, demonstrating preservation of key properties and developing related theoretical frameworks.

Contribution

It introduces the nonstandard hull construction for C*-algebras and proves that several important properties are preserved and reflected, extending the theory of weights.

Findings

Preservation of being infinite and properly infinite C*-algebras

Preservation of generation by mutually orthogonal projections

Development of a theory of weights in nonstandard hulls

Abstract

We study properties of C*-algebras obtained from the nonstandard hull construction (a generalization of the ultraproduct of C*-algebras). Among others, we prove that the properties of being an infinite and a properly infinite C*-algebra are both preserved and reflected by the nonstandard hull construction. We also show that the property of being generated by mutually orthogonal projections is preserved by the nonstandard hull. Furthermore, we indicate how a theory of weights can be developed in a nonstandard hull. Eventually we provide a few examples of standard results that follow from the nonstandard results previously obtained.