UHF slicing and classification of nuclear C*-algebras

TL;DR

This paper demonstrates that certain simple RSH C*-algebras become tracially approximately interval algebras after tensoring with a UHF algebra, enabling their classification via Elliott invariants.

Contribution

It introduces a method to classify specific RSH C*-algebras by showing they are tracially approximately interval after tensoring with a UHF algebra, relaxing previous assumptions.

Findings

C*-algebras are classifiable by Elliott invariants

Established a path resembling a discrete [0,1] interval within the algebra

Extended classification methods to include examples of minimal dynamical systems

Abstract

In this paper we show that certain simple locally recursive subhomogeneous (RSH) C*-algebras are tracially approximately interval algebras after tensoring with the universal UHF algebra. This involves a linear algebraic encoding of the structure of the local RSH algebra allowing us to find a path through the algebra which looks like a discrete version of [0,1] and exhausts most of the algebra. We produce an actual copy of the interval and use properties of C*-algebras tensored with UHF algebras to move the honest interval underneath the discrete version. It follows from our main result that such C*-algebras are classifiable by Elliott invariants. Our theorem requires finitely many tracial states that all induce the same state on the K_0-group; in particular we do not require that projections separate tracial states. We apply our results to classify some examples of C*-algebras constructed by Elliott to exhaust the invariant. We also give an alternate way to classify examples of Lin and Matui of C*-algebras of minimal dynamical systems. In this way our result can be viewed as a first step towards removing the requirement that projections separate tracial states in the classification theorem for C*-algebras of minimal dynamical systems given by Toms and the second named author.