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Mackey Imprimitivity and commuting tuples of homogeneous normal operators

Gadadhar Misra, E. K. Narayanan, Cherian Varughese

Abstract

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting $d$- tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a $*$- algebra representation $ρ$ of $C_0(\mathbb{S})$ (where $\mathbb S$ is a locally compact Hausdorff space) into a direct sum. If there is a group $G$ acting transitively on $\mathbb{S}$ and is adapted to the $*$- representation $ρ$ via a unitary representation $U$ of the group $G$, in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation $U$ becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space $S\subset \mathbb {C}^d$ decomposes into finitely many $G$- orbits. In such cases, the imprimitivity based on $S$ admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of $G$- orbits.

Mackey Imprimitivity and commuting tuples of homogeneous normal operators

Abstract

In this semi-expository article, we investigate the relationship between the imprimitivity introduced by Mackey several decades ago and commuting - tuples of homogeneous normal operators. The Hahn-Hellinger theorem gives a canonical decomposition of a - algebra representation of (where is a locally compact Hausdorff space) into a direct sum. If there is a group acting transitively on and is adapted to the - representation via a unitary representation of the group , in other words, if there is an imprimitivity, then the Hahn-Hellinger decomposition reduces to just one component, and the group representation becomes an induced representation, which is Mackey's imprimitivity theorem. We consider the case where a compact topological space decomposes into finitely many - orbits. In such cases, the imprimitivity based on admits a decomposition as a direct sum of imprimitivities based on these orbits. This decomposition leads to a correspondence with homogeneous normal tuples whose joint spectrum is precisely the closure of - orbits.
Paper Structure (6 sections, 14 theorems, 51 equations)

This paper contains 6 sections, 14 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Spectral theorem and the Hahn-Hellinger decomposition
  3. Multiplier representations and Induced representations
  4. Multiplier representations
  5. Induced representation
  6. Imprimitivity and homogeneous normal operators

Key Result

Theorem 2.2

Suppose that $\mathbb{S}$ is a locally compact Hausdorff space, and $\rho$ is a nondegenerate $*$- representation of $C_0(\mathbb{S})$ on $\mathcal{H}$. Then there is a unique regular projection-valued measure $P$ on $\mathbb{S}$ such that $\rho(f)=\int f d P$ for all $f \in C_0(\mathbb{S})$.

Theorems & Definitions (31)

  • Definition 2.1
  • Theorem 2.2: Corollary 1.55, GF
  • Theorem 2.3: Chapter IX, Theorem 1.14, JBC
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Theorem 2.8: Theorem 3, Hastings
  • Lemma 2.9
  • ...and 21 more