Real rank of extensions of $C^*$-algebras
Volume 276 / 2024
Studia Mathematica 276 (2024), 131-155
MSC: Primary 46L05; Secondary 46L45, 46L85, 46M20
DOI: 10.4064/sm231119-19-4
Published online: 5 June 2024
Abstract
Given a closed ideal $A$ in a $C^*$-algebra $E$, we develop techniques to bound the real rank of $E$ in terms of the real ranks of $A$ and $E/A$. Building on work of Brown, Lin and Zhang, we obtain explicit computations if $A$ belongs to any of the following classes: (1) $C^*$-algebras with real rank zero, stable rank one and vanishing $K_1$-group; (2) simple, purely infinite $C^*$-algebras; (3) simple, $\mathcal Z$-stable $C^*$-algebras with real rank zero; (4) separable, stable $C^*$-algebras with an approximate unit of projections and the Corona Factorization Property.
