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Commutative C* algebras and Gelfand theory through phase space methods

Robert Fulsche, Oliver Fürst

Abstract

We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of quantum spectral synthesis, which were recently used to characterize certain commutative Toeplitz algebras on the Fock space. Our method applies to this setting and also to more general abelian phase spaces. Besides characterizing Gelfand spectra of such commutative operator algebras, we also prove an extension of this result to the operator-valued case.

Commutative C* algebras and Gelfand theory through phase space methods

Abstract

We show how the Gelfand spectrum of certain commutative operator algebras can be studied based on the theorem of Stone and von Neumann. The method presented is a natural addition to the tools of quantum spectral synthesis, which were recently used to characterize certain commutative Toeplitz algebras on the Fock space. Our method applies to this setting and also to more general abelian phase spaces. Besides characterizing Gelfand spectra of such commutative operator algebras, we also prove an extension of this result to the operator-valued case.

Paper Structure

This paper contains 4 sections, 19 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. An operator-valued version of the theorem
  4. Examples

Key Result

Theorem 1.2

Baggett_Kleppner1973 Let $(\Xi, m)$ be a phase space. Then, there exists a unique (up to unitary equivalence) irreducible projective unitary representation $(\mathcal{H}, U)$ with $m$ as its multiplier, i.e., $U_x U_y = m(x,y) U_{x+y}$ for all $x, y \in \Xi$.

Theorems & Definitions (37)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 27 more