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Spectral finiteness, quantum norm continuity and classical points

Alexandru Chirvasitu

Abstract

We prove various notions of uniform continuity for compact-quantum-group representations on Hilbert or Banach spaces equivalent to having finite spectrum, i.e. finitely many isotypic components. This generalizes the classical analogue for compact-group representations on Banach spaces, and relies in part on Riemann-Lebesgue-type decay properties for Fourier coefficients of elements in minimal tensor products with compact-quantum-group function algebras.

Spectral finiteness, quantum norm continuity and classical points

Abstract

We prove various notions of uniform continuity for compact-quantum-group representations on Hilbert or Banach spaces equivalent to having finite spectrum, i.e. finitely many isotypic components. This generalizes the classical analogue for compact-group representations on Banach spaces, and relies in part on Riemann-Lebesgue-type decay properties for Fourier coefficients of elements in minimal tensor products with compact-quantum-group function algebras.
Paper Structure (1 section, 10 theorems, 42 equations)

This paper contains 1 section, 10 theorems, 42 equations.

Table of Contents

  1. Quantum uniform continuity in several guises, and consequences

Key Result

Theorem 1

The following conditions on a unitary ${\mathbb G}$-representation $U\in M({\mathcal{C}}({\mathbb G})\underline{\otimes}{\mathcal{K}}(H))$ on a Hilbert space $H$ are equivalent.

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1.1
  • Proposition 1.2
  • Proof 1
  • Definition 1.3
  • Lemma 1.4
  • Proof 2
  • Definition 1.5
  • ...and 9 more