From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero

TL;DR

This paper establishes a connection between stable rank one and real rank zero in simple separable C*-algebras using tracial oscillation, showing that certain algebraic properties imply real rank zero in associated sequence algebras.

Contribution

It demonstrates that stable rank one implies tracial approximate oscillation zero and that this condition ensures the associated sequence algebra has real rank zero.

Findings

Stable rank one implies tracial approximate oscillation zero.

Tracial approximate oscillation zero leads to real rank zero in the sequence algebra.

Equivalence of tracial approximate oscillation zero and real rank zero in algebras with non-trivial 2-quasitraces.

Abstract

We present a relation between stable rank one and real rank zero via the method of tracial oscillation. Let $A$ be a simple separable $C^*$-algebra of stable rank one. We show that $A$ has tracial approximate oscillation zero and, as a consequence, the tracial sequence algebra $l^\infty(A)/J_A$ has real rank zero, where $J_A$ is the trace-kernel ideal with respect to 2-quasitraces. We also show that for a $C^*$-algebra $B$ that has non-trivial 2-quasitraces, $B$ has tracial approximate oscillation zero is equivalent to $l^\infty(B)/J_B$ has real rank zero.