Homotopical Stable Ranks for Certain $C^{\ast}$-algebras Associated to Groups

TL;DR

This paper investigates the homotopical stable ranks of specific $C^{ ext{*}}$-algebras, including those associated with groups, $C(X)$-algebras, and crossed products, providing new estimates for these ranks.

Contribution

It introduces new estimates for the stable ranks of group $C^{ ext{*}}$-algebras and crossed product $C^{ ext{*}}$-algebras with finite group actions, expanding understanding of their homotopical properties.

Findings

Estimated stable ranks for certain $C(X)$-algebras.

Derived bounds for group $C^{ ext{*}}$-algebras.

Provided estimates for crossed product $C^{ ext{*}}$-algebras with Rokhlin property.

Abstract

We study the general and connected stable ranks for $C^{\ast}$-algebras. We estimate these ranks for certain $C(X)$-algebras, and use that to do the same for certain group $C^{\ast}$-algebras. Furthermore, we also give estimates for the ranks of crossed product $C^{\ast}$-algebras by finite group actions with the Rokhlin property.