From Stable Rank One to Real Rank Zero: A Note on Tracial Approximate Oscillation Zero
Xuanlong Fu
TL;DR
The paper investigates how stable rank one regularity forces tracial regularity by introducing tracial approximate oscillation zero (TAOZ) and showing it implies real rank zero for the tracial sequence algebra $l_ ty(A)/J_A$. Building a bridge via hereditary surjective canonical maps, it combines Antoine-Perera-Robert-Thiel results with Lin's techniques to prove that separable simple $C^*$-algebras of stable rank one have TAOZ, and hence $l_ ty(A)/J_A$ has real rank zero; this also yields an equivalence: for any $B$ with nontrivial $2$-quasitraces, TAOZ for $B$ is equivalent to $l_ ty(B)/J_B$ having real rank zero. The results apply to diagonal AH algebras and $Z^d$-crossed products, broadening the class of algebras for which the Elliott invariant and tracial data control the regularity, in line with the Toms–Winter program. Overall, the work advances understanding of when stable rank one implies real rank zero in a tracial sense and clarifies the role of trace-based regularity in classification contexts.
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.