Diagonalizing matrices over AW*-algebras

TL;DR

This paper proves that commuting normal matrices over AW*-algebras can be simultaneously diagonalized, using a new dimension theory for projections, and shows that matrix ring formation is a functor in this context.

Contribution

It introduces a dimension theory for properly infinite projections in AW*-algebras and demonstrates that matrix rings form a functor on the category of AW*-algebras.

Findings

Commuting normal matrices over AW*-algebras can be simultaneously diagonalized.

A new dimension theory for properly infinite projections is developed.

Passing to matrix rings is a functor on the category of AW*-algebras.

Abstract

Every commuting set of normal matrices with entries in an AW*-algebra can be simultaneously diagonalized. To establish this, a dimension theory for properly infinite projections in AW*-algebras is developed. As a consequence, passing to matrix rings is a functor on the category of AW*-algebras.