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Projective representations on operator algebras

Sergio Girón Pacheco

Abstract

We establish new restrictions on the values of the lifting obstruction for projective unitary representations of second countable, locally compact Hausdorff groups on operator algebras. Using these, we show that every projective representation on the Jiang--Su algebra lifts to a genuine group representation. We also characterise the possible values of lifting obstructions for finite group projective representations on UHF-algebras of infinite type and Cuntz algebras. Finally we show that certain 2-cocycles of Property (T) groups cannot arise as lifting invariants of projective representations on the Hyperfinite II$_{1}$ factor.

Projective representations on operator algebras

Abstract

We establish new restrictions on the values of the lifting obstruction for projective unitary representations of second countable, locally compact Hausdorff groups on operator algebras. Using these, we show that every projective representation on the Jiang--Su algebra lifts to a genuine group representation. We also characterise the possible values of lifting obstructions for finite group projective representations on UHF-algebras of infinite type and Cuntz algebras. Finally we show that certain 2-cocycles of Property (T) groups cannot arise as lifting invariants of projective representations on the Hyperfinite II factor.

Paper Structure

This paper contains 4 sections, 12 theorems, 42 equations.

Table of Contents

  1. Preliminaries
  2. Projective unitary representations
  3. projective representations on C*-algebras
  4. A 2-cocycle that does not arise on $\mathcal{R}$

Key Result

Theorem 1

Every projective representation of a separable locally compact group $G$ on the Jiang--Su algebra $\mathcal{Z}$ lifts to a genuine representation of $G$.

Theorems & Definitions (26)

  • Theorem : cf. Theorem \ref{['thm:JiangSu']}
  • Theorem : cf. Theorem \ref{['thm:UHF']} and Theorem \ref{['thm:cuntz']}
  • Lemma 1.1: cf. MO76
  • Lemma 1.2: cf. MO76
  • Definition 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 16 more