Stable rank for inclusions of C*-algebras

TL;DR

This paper establishes an upper bound of 2 for the topological stable rank of certain unital C*-algebras containing a simple subalgebra with the positive spectral property, especially under group actions.

Contribution

It proves that if a unital C*-algebra contains a simple subalgebra with the positive spectral property and bounded stable ranks of corner subalgebras, then the algebra's stable rank is at most 2, extending previous results.

Findings

If a unital C*-algebra has a simple subalgebra with ext{ extbf{PSP}} and bounded corner stable ranks, then its stable rank is at most 2.

Group actions on a C*-algebra with stable rank one and ext{ extbf{PSP}} preserve the stable rank bound of 2.

The result applies to iterated crossed products by finite groups, ensuring their stable rank does not exceed 2.

Abstract

When a unital \ca $A$ has topological stable rank one (write $\tsr(A) = 1$), we know that $\tsr(pAp) \leq 1$ for a non-zero projection $p \in A$. When, however, $\tsr(A) \geq 2$, it is generally faluse. We prove that if a unital C*-algebra $A$ has a simple unital C*-subalgebra $D$ of $A$ with common unit such that $D$ has \PSP and $\sup_{p\in P(D)}\tsr(pAp) < \infty$, then $\tsr(A) \leq 2.$ As an application let $A$ be a simple unital \ca with $\tsr(A) = 1$ and \PSP, $\{G_k\}_{k=1}^n$ finite groups, $\af_k$ actions from $G_k$ to ${\rm Aut}((...((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_{k-1}}G_{k-1}).$ $(G_0 = \{1\})$ Then $$ \tsr((... ((A\times_{\af_1}G_1)\times_{\af_2} G_2)...)\times_{\af_n}G_n) \leq 2. $$