On the Banach space isomorphism type of AF C*-algebras and their triangular subalgebras

TL;DR

This paper proves that all non-Type I AF C*-algebras are Banach space isomorphic, extending previous results, and explores similar classifications for their triangular subalgebras and continua.

Contribution

It generalizes the Banach space isomorphism classification to all non-Type I AF C*-algebras and their subalgebras, expanding prior work by Arazy.

Findings

Non-Type I AF C*-algebras are Banach space isomorphic

Results extend to triangular subalgebras of AF C*-algebras

Discussion on continua of Type I AF C*-algebras

Abstract

It is shown that all the approximately finite dimensional C*-algebras which are not of Type I are isomorphic as Banach spaces. This generalises the matroid case given previously by Arazy. Analogous results are obtained for various families of triangular subalgebras of AF C*-algebras. In addition the classification of various continua of Type I AF C*-algebras is discussed.