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Real rank of some multiplier algebras

Hannes Thiel

Abstract

We show that there exists a separable, nuclear C*-algebra with real rank zero and trivial K-theory such that its multiplier and corona algebra have real rank one. This disproves two conjectures of Brown and Pedersen. We also compute the real rank of the stable multiplier algebra and the stable corona algebra of countably decomposable type $\mathrm{I}_\infty$ and type $\mathrm{II}_\infty$ factors. Together with results of Zhang this completes the computation of the real rank for stable multiplier and corona algebras of countably decomposable factors.

Real rank of some multiplier algebras

Abstract

We show that there exists a separable, nuclear C*-algebra with real rank zero and trivial K-theory such that its multiplier and corona algebra have real rank one. This disproves two conjectures of Brown and Pedersen. We also compute the real rank of the stable multiplier algebra and the stable corona algebra of countably decomposable type and type factors. Together with results of Zhang this completes the computation of the real rank for stable multiplier and corona algebras of countably decomposable factors.
Paper Structure (6 sections, 24 theorems, 91 equations)

This paper contains 6 sections, 24 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on real rank of extensions
  3. Real rank of multiplier algebras of some simple C*-algebras
  4. Extensions by the algebra of compact operators
  5. Extensions by simple, purely infinite C*-algebras
  6. Extensions by certain simple C*-algebras with stable rank one

Key Result

Theorem 1

Let $N$ be a countably decomposable factor. If $N$ has type $\mathrm{I}_\infty$ or type $\mathrm{II}_\infty$, then If $N$ has type $\mathrm{II}_1$ or type $\mathrm{III}$, then

Theorems & Definitions (56)

  • Example 1: \ref{['exa:SeparableBPCounterexample']}
  • Theorem 1: \ref{['prp:rrStableMultBdd']}, \ref{['prp:rrStableMultIIinf']}
  • Theorem 2: \ref{['prp:rrMultExtByKK']}, \ref{['prp:rrMultExtSimplePI']}, \ref{['prp:rrMultExtSimpleSR1']}
  • Definition 1: Brown-Pedersen BroPed91CAlgRR0
  • Definition 2: Thi23arX:RRExt
  • Theorem 1: Thi23arX:RRExt
  • Theorem 2: Thi23arX:RRExt
  • Proposition 1: Thi23arX:RRExt
  • Proposition 2: Thi23arX:RRExt
  • Proposition 3
  • ...and 46 more