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Operator algebras and unitary representations of Polish groups

Raul Quiroga-Barranco

Abstract

We study the relationship between operator algebras, $C^*$ and von Neumann, acting on a Hilbert space and unitary representations of topological groups on the same space. We obtain certain correspondences between both these families of objects. In particular, we prove that the study of Abelian subalgebras of a norm-separable $C^*$-algebra can be, in many aspects, reduced to the study of unitary representations of Polish groups. It is shown that our techniques have applications to the construction and classification of Abelian subalgebras of operator $C^*$-algebras on separable Hilbert spaces.

Operator algebras and unitary representations of Polish groups

Abstract

We study the relationship between operator algebras, and von Neumann, acting on a Hilbert space and unitary representations of topological groups on the same space. We obtain certain correspondences between both these families of objects. In particular, we prove that the study of Abelian subalgebras of a norm-separable -algebra can be, in many aspects, reduced to the study of unitary representations of Polish groups. It is shown that our techniques have applications to the construction and classification of Abelian subalgebras of operator -algebras on separable Hilbert spaces.
Paper Structure (14 sections, 25 theorems, 41 equations)

This paper contains 14 sections, 25 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Operator algebras and unitary groups
  3. Some facts and conventions
  4. The unitary group of an operator algebra
  5. Operator $C^*$-algebras and topological groups
  6. Intertwining maps of topological groups
  7. Operator $C^*$-algebras as intertwining maps (topological groups)
  8. Decompositions of spaces and algebras
  9. Direct integrals of Hilbert spaces
  10. Decompositions of operator algebras
  11. Operator $C^*$-algebras and Polish groups
  12. Operator $C^*$-algebras as intertwining operators (Polish groups)
  13. Norm-separable and Abelian operator $C^*$-algebras
  14. Compact groups vs Polish groups

Key Result

Theorem 1

Let $\mathcal{H}$ be a separable Hilbert space and let $\mathfrak{A} \subset \mathcal{B}(\mathcal{H})$ be an Abelian norm-separable $C^*$-subalgebra of $\mathcal{B}(\mathcal{H})$. Then, there exist a Polish norm-closed subgroup $G \subset \mathrm{U}(\mathcal{H})_{\mathrm{N}}$ such that with respect

Theorems & Definitions (63)

  • Theorem : Decomposition Theorem for Abelian $C^*$-Algebras
  • Proposition 2.2.1
  • Proposition 2.2.2
  • Proposition 2.2.3
  • Theorem 3.2.1
  • proof
  • Remark 3.2.2
  • Remark 3.2.3
  • Corollary 3.2.4
  • proof
  • ...and 53 more