Decomposition rank of UHF-absorbing C*-algebras

TL;DR

This paper establishes that certain nuclear, quasidiagonal, simple C*-algebras have low decomposition rank when tensoring with UHF-algebras, providing new insights into their structure and counterexamples to existing conjectures.

Contribution

It proves that nuclear, quasidiagonal, simple C*-algebras with strict comparison have finite decomposition rank and offers a new proof for their tracial rank zero when tensoring with UHF-algebras.

Findings

A unital simple C*-algebra with a unique trace tensoring with UHF has decomposition rank at most one.

Nuclear, quasidiagonal, strict comparison C*-algebras have finite decomposition rank.

Counter-example to the Powers-Sakai conjecture is provided.

Abstract

Let A be a unital separable simple C*-algebra with a unique tracial state. We prove that if A is nuclear and quasidiagonal, then A tensored with the universal UHF-algebra has decomposition rank at most one. Then it is proved that A is nuclear, quasidiagonal and has strict comparison if and only if A has finite decomposition rank. For such A, we also give a direct proof that A tensored with a UHF-algebra has tracial rank zero. Applying this characterization, we obtain a counter-example to the Powers-Sakai conjecture.