C*-Algebras with the Approximate Positive Factorization Property

TL;DR

This paper systematically studies C*-algebras with the approximate positive factorization property (APFP), revealing their structural properties, passing of APFP to subalgebras, and providing new examples including type II_1 factors and certain direct limits.

Contribution

It offers a comprehensive analysis of APFP in C*-algebras, establishing key properties, stability under algebraic operations, and introducing a new concept of rank for these algebras.

Findings

APFP algebras have connected invertible groups and trivial K_1.

APFP passes to matrix algebras and certain extensions.

New examples include type II_1 factors and infinite-dimensional simple limits.

Abstract

We say that a unital C*-algrebra A has the approximate positive factorization property (APFP) if every element of A is a norm limit of products of positive elements of A. (There is also a definition for the nonunital case.) T. Quinn has recently shown that a unital AF algebra has the APFP if and only if it has no finite dimensional quotients. This paper is a more systematic investigation of C*-algebras with the APFP. We prove various properties of such algebras. For example: They have connected invertible group, trivial K_1, and stable rank 1. In the unital case, the K_0 group separates the tracial states. The APFP passes to matrix algebras. and if I is an ideal in A such that I and A/I have the APFP, then so does A. We also give some new examples of C*-algebras with the APFP, including type II_1 factors and infinite-dimensional simple unital direct limits with slow dimension growth, real rank zero, and trivial K_1 group. An infinite- dimensional simple unital direct limit with slow dimension growth and with the APFP must have real rank zero, but we also give examples of unital algebras with the APFP which do not have real rank zero. Our analysis also leads to the introduction of a new concept of rank for a C*-algebra that may be of interest in the future.