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Real rank of extensions of C*-algebras

Hannes Thiel

Abstract

Given a closed ideal $I$ in a C*-algebra $A$, we develop techniques to bound the real rank of $A$ in terms of the real ranks of $I$ and $A/I$. Building on work of Brown, Lin and Zhang, we obtain complete solutions if $I$ 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.

Real rank of extensions of C*-algebras

Abstract

Given a closed ideal in a C*-algebra , we develop techniques to bound the real rank of in terms of the real ranks of and . Building on work of Brown, Lin and Zhang, we obtain complete solutions if belongs to any of the following classes: (1) C*-algebras with real rank zero, stable rank one and vanishing -group; (2) simple, purely infinite C*-algebras; (3) simple, -stable C*-algebras with real rank zero; (4) separable, stable C*-algebras with an approximate unit of projections and the Corona Factorization Property.
Paper Structure (5 sections, 25 theorems, 67 equations)

This paper contains 5 sections, 25 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. The real rank of extensions
  3. The extension real rank
  4. Permanence properties of the extension real rank
  5. C*-algebras with low extension real rank

Key Result

Theorem 1

Given an extension $0 \to A \to E \to B\to 0$ of $C^*$-algebras, we have

Theorems & Definitions (60)

  • Theorem 1: \ref{['prp:rrExt']}
  • Theorem 2: \ref{['prp:rrExt-xrr-rr']}
  • Definition 1: Brown-Pedersen, BroPed91CAlgRR0
  • Remark 1
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Proposition 2: LinRor95ExtLimitCircle
  • proof
  • ...and 50 more