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Cartan subalgebras in W*-algebras

Jean Renault

Abstract

This article presents a proof of the Feldman-Moore theorem on Cartan subalgebras in W*-algebras based on the non-commutative Stone equivalence between Boolean inverse semigroups and Boolean groupoids. The proof is decomposed into two parts. The first part writes the W*-algebra as the W*-algebra of the Weyl twist of the Cartan subalgebra. The second part is an extension to separable measure inverse semigroups of the realization of a separable measure algebra by a standard measure space.

Cartan subalgebras in W*-algebras

Abstract

This article presents a proof of the Feldman-Moore theorem on Cartan subalgebras in W*-algebras based on the non-commutative Stone equivalence between Boolean inverse semigroups and Boolean groupoids. The proof is decomposed into two parts. The first part writes the W*-algebra as the W*-algebra of the Weyl twist of the Cartan subalgebra. The second part is an extension to separable measure inverse semigroups of the realization of a separable measure algebra by a standard measure space.
Paper Structure (8 sections, 36 theorems, 64 equations)

This paper contains 8 sections, 36 theorems, 64 equations.

Table of Contents

  1. Twisted Boolean inverse semigroups and groupoids
  2. Boolean inverse semigroups and Boolean groupoids
  3. Twists
  4. Convolution algebras
  5. Abstract realization of a Cartan pair
  6. Concrete realization of a Cartan pair
  7. Non-commutative Stone equivalence
  8. Representations of Boolean inverse semigroups

Key Result

Lemma 1.9

Let $(G,\Sigma)$ be a twisted Boolean groupoid. For $S$ and $T$ in ${\rm Bis}(\Sigma))$, we have $\Delta_S*\Delta_T=\Delta_{ST}$, $\Delta_S^*=\Delta_{S^{-1}}$. If $S$ belongs to ${\mathcal{T}}(G^{(0)})$ and is given by a function $b$, then $\Delta_S=\tilde{b}$ as defined above.

Theorems & Definitions (112)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Lemma 1.9
  • Lemma 1.10
  • ...and 102 more