Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

(Quantum) discreteness, spectrum compactness and uniform continuity

Alexandru Chirvasitu

TL;DR

The paper investigates when actions of compact (quantum) groups on Banach spaces are uniformly continuous and how this relates to finiteness of the spectrum and extensibility to multiplier algebras. It develops a uniformity framework for Compact Quantum Group actions, extends spectral decompositions to Banach spaces, and establishes that finite isotypic spectrum is equivalent to uniform continuity, with further connections to classical compact groups via subgroup restrictions. A key discrete-structure result shows discreteness of locally compact groups is equivalent to action-coproduct preservation across Stone–Čech compactifications and related forgetful functors in both topological and operator-algebraic settings. Finally, the work introduces jointly monic families of quantum subgroups and proves a lifting criterion for linearly reductive quantum groups: a representation has finitely many isotypic components iff its restrictions to a normal subgroup and a topologically generating subgroup do so, clarifying how local spectral data governs global uniformity.

Abstract

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions. Examples include: (a) a unitary representation of a compact quantum group induces a continuous action on the $C^*$-algebra of bounded operators if and only if it has finitely many isotypic components, and hence is uniformly continuous; (b) a compact quantum group is finite if and only if its continuous actions on $C^*$-algebras lift to continuous actions on either the multiplier algebras or von Neumann envelopes thereof; (c) a (classical) locally compact group $\mathbb{G}$ is discrete if and only if the forgetful functor from $\mathbb{G}$-acted-upon compact $T_2$ spaces back to compact $T_2$ spaces creates coproducts; (d) a representation of a linearly reductive quantum group has finitely many isotypic components if and only if its restrictions to two topologically-generating quantum subgroups, one of which is normal, do; (e) equivalent characterizations of uniform continuity for actions of compact groups on Banach spaces, e.g. that such an action is uniformly continuous if and only if its restrictions to a pro-torus and to pro-$p$ subgroups are.

(Quantum) discreteness, spectrum compactness and uniform continuity

TL;DR

The paper investigates when actions of compact (quantum) groups on Banach spaces are uniformly continuous and how this relates to finiteness of the spectrum and extensibility to multiplier algebras. It develops a uniformity framework for Compact Quantum Group actions, extends spectral decompositions to Banach spaces, and establishes that finite isotypic spectrum is equivalent to uniform continuity, with further connections to classical compact groups via subgroup restrictions. A key discrete-structure result shows discreteness of locally compact groups is equivalent to action-coproduct preservation across Stone–Čech compactifications and related forgetful functors in both topological and operator-algebraic settings. Finally, the work introduces jointly monic families of quantum subgroups and proves a lifting criterion for linearly reductive quantum groups: a representation has finitely many isotypic components iff its restrictions to a normal subgroup and a topologically generating subgroup do so, clarifying how local spectral data governs global uniformity.

Abstract

We prove a number of results linking properties of actions by compact groups (both quantum and classical) on Banach spaces, such as uniform continuity, spectrum finiteness and extensibility of the actions across several constructions. Examples include: (a) a unitary representation of a compact quantum group induces a continuous action on the -algebra of bounded operators if and only if it has finitely many isotypic components, and hence is uniformly continuous; (b) a compact quantum group is finite if and only if its continuous actions on -algebras lift to continuous actions on either the multiplier algebras or von Neumann envelopes thereof; (c) a (classical) locally compact group is discrete if and only if the forgetful functor from -acted-upon compact spaces back to compact spaces creates coproducts; (d) a representation of a linearly reductive quantum group has finitely many isotypic components if and only if its restrictions to two topologically-generating quantum subgroups, one of which is normal, do; (e) equivalent characterizations of uniform continuity for actions of compact groups on Banach spaces, e.g. that such an action is uniformly continuous if and only if its restrictions to a pro-torus and to pro- subgroups are.
Paper Structure (7 sections, 18 theorems, 49 equations)

This paper contains 7 sections, 18 theorems, 49 equations.

Table of Contents

  1. Preliminaries
  2. Uniform CQG actions on Banach spaces
  3. Generalities on representations and uniformity
  4. Plain compact groups
  5. Returning to quantum groups
  6. Discreteness as action-coproduct preservation
  7. Jointly monic families and uniformity

Key Result

Theorem 1

A unitary representation of a compact quantum group ${\mathbb G}$ on a Hilbert space $H$ induces a continuous conjugation action on ${\mathcal{L}}(H)$ if and only if $\rho$ has finitely many isotypic components. In particular, said action on ${\mathcal{L}}(H)$ will then automatically be uniformly c

Theorems & Definitions (37)

  • Definition 1
  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 2.4
  • Proof 1
  • ...and 27 more