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Essential Cartan subalgebras of $C^*$-algebras

Jonathan Taylor

TL;DR

The paper extends Renault's Cartan reconstruction to essential Cartan pairs, replacing conditional expectations with local multiplier algebra-valued pseudoexpectations and allowing non-Hausdorff, non-separable, and aperiodic settings. It shows that every essential Cartan pair $(A,B)$ arises from an essential twisted groupoid C*-algebra $C^*_{\mathrm{ess}}(G(A,B),\Sigma(A,B))$, with $A\cong C_0(G^{(0)})$ and $B\cong C^*_{\mathrm{ess}}(G(A,B),\Sigma(A,B))$, and that the Weyl twist is effective and unique up to isomorphism among twists on effective groupoids. The work develops a generalized evaluation framework to represent normalisers as elements of a dense subalgebra, establishes a robust relation between normalisers, partial homeomorphisms, and the Weyl groupoid, and proves that automorphisms of twists correspond precisely to Cartan automorphisms when the base groupoid is effective. Overall, it strengthens the bridge between groupoid dynamics and noncommutative Cartan theory, with applications to classification-type questions and the study of regular masa inclusions beyond separable or Hausdorff contexts.

Abstract

We define essential commutative Cartan pairs of $C^*$-algebras generalising the definition of Renault and show that such pairs are given by essential twisted groupoid $C^*$-algebras as defined by Kwaśniewski and Meyer. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential commutative Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.

Essential Cartan subalgebras of $C^*$-algebras

TL;DR

The paper extends Renault's Cartan reconstruction to essential Cartan pairs, replacing conditional expectations with local multiplier algebra-valued pseudoexpectations and allowing non-Hausdorff, non-separable, and aperiodic settings. It shows that every essential Cartan pair arises from an essential twisted groupoid C*-algebra , with and , and that the Weyl twist is effective and unique up to isomorphism among twists on effective groupoids. The work develops a generalized evaluation framework to represent normalisers as elements of a dense subalgebra, establishes a robust relation between normalisers, partial homeomorphisms, and the Weyl groupoid, and proves that automorphisms of twists correspond precisely to Cartan automorphisms when the base groupoid is effective. Overall, it strengthens the bridge between groupoid dynamics and noncommutative Cartan theory, with applications to classification-type questions and the study of regular masa inclusions beyond separable or Hausdorff contexts.

Abstract

We define essential commutative Cartan pairs of -algebras generalising the definition of Renault and show that such pairs are given by essential twisted groupoid -algebras as defined by Kwaśniewski and Meyer. We show that the underlying twisted groupoid is effective, and is unique up to isomorphism among twists over effective groupoids giving rise to the essential commutative Cartan pair. We also show that for twists over effective groupoids giving rise to such pairs, the automorphism group of the twist is isomorphic to the automorphism group of the induced essential Cartan pair via explicit constructions.
Paper Structure (10 sections, 44 theorems, 60 equations)

This paper contains 10 sections, 44 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. $C^*$-algebras of twists over groupoids
  3. The essential $C^*$-algebra
  4. Aperiodic and Masa inclusions
  5. Essential Cartan subalgebras of $C^*$-algebras
  6. The evaluation map
  7. Uniqueness of the Weyl groupoid and twist
  8. Non-uniqueness for non-effective groupoids
  9. More general regular masa inclusions
  10. Automorphisms of twists and Cartan automorphisms

Key Result

Theorem 1

Let $A\subseteq B$ be an essential Cartan pair with faithful local conditional expectation $E:B\to M_{\mathrm{loc}}(A)$. Let $(G(A,B),\Sigma(A,B))$ by the Weyl twist associated to $A\subseteq B$. There is a ${}^*$-isomorphism $C^*_\mathrm{ess}(G(A,B),\Sigma(A,B))\cong B$ that restricts to an isomorp

Theorems & Definitions (98)

  • Theorem 1: see Theorem \ref{['thm-isoEssTwstGrpdCstarAlg']}
  • Definition 2.1: Kumjian_CstarDiags
  • Proposition 2.2: Kumjian_CstarDiags
  • Lemma 2.3: Kumjian_CstarDiags
  • Corollary 2.4
  • proof
  • Definition 2.5: Renault_Cartan
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 88 more