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Uniformity and isotypic smallness for quantum-group representations

Alexandru Chirvasitu

Abstract

Compact-group representations on Banach spaces are known to be norm-continuous precisely when they have finite spectra. For a quantum group with continuous-function algebra $\mathcal{C}(\mathbb{G})$ norm continuity can be cast analogously as the bounded weak$^*$-norm continuity of the representation's attached map $\mathcal{C}(\mathbb{G})^*\to \mathrm{End}(E)$. While the uniformity/isotypic finiteness equivalence no longer holds generally, it does for compact quantum groups either coamenable or having dimension-bounded irreducible representations. This generalizes the aforementioned classical variant, providing two independent quantum-specific mechanisms of recovering it.

Uniformity and isotypic smallness for quantum-group representations

Abstract

Compact-group representations on Banach spaces are known to be norm-continuous precisely when they have finite spectra. For a quantum group with continuous-function algebra norm continuity can be cast analogously as the bounded weak-norm continuity of the representation's attached map . While the uniformity/isotypic finiteness equivalence no longer holds generally, it does for compact quantum groups either coamenable or having dimension-bounded irreducible representations. This generalizes the aforementioned classical variant, providing two independent quantum-specific mechanisms of recovering it.

Paper Structure

This paper contains 1 section, 6 theorems, 10 equations.

Table of Contents

  1. Controlled matrix-coefficient decay and its bearing on uniformity

Key Result

Theorem 1

Banach-space representations of coamenable compact quantum groups are uniform$_{\le 1}$ precisely when they have finitely many isotypic components. $\blacksquare$

Theorems & Definitions (12)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 1.4
  • Proof 1
  • Lemma 1.5
  • Remark 1.6
  • Definition 1.7
  • ...and 2 more