$C^*$-Algebras with the Approximate Positive Factorization Property

Abstract:

We say that a unital $\mathrm {C}^{*}$-algebra $A$ 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 $\mathrm {C}^{*}$-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 $\mathrm {C}^{*}$-algebras with the APFP, including type $\mathrm {II}_{1}$ factors and infinite-dimensional simple unital direct limits of homogeneous $\mathrm {C}^{*}$-algebras with slow dimension growth, real rank zero, and trivial $K_{1}$ group. Simple direct limits of homogeneous $\mathrm {C}^{*}$-algebras with slow dimension growth which have the APFP must have real rank zero, but we also give examples of (nonsimple) unital algebras with the APFP which do not have real rank zero. Our analysis leads to the introduction of a new concept of rank for a $\mathrm {C}^{*}$-algebra that may be of interest in the future.