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Twisted topological correspondences and Cartan subalgebras

Aaron Kettner

TL;DR

This work introduces twisted topological correspondences as a unifying geometric framework for objects generalizing Katsura’s topological correspondences and Li’s twisted graphs. It proves a bijection between twisted correspondences and principal bundles valued in unitary normalizers, and shows how these structures yield $C^*$-correspondences with two equivalent characterizations: (i) atlases with $\\mathcal{U}(N_{D_n})$-valued transitions compatible with the left action, and (ii) Cartan subalgebras in the compacts that contain the left action; these viewpoints lead to rigidity results. A central outcome is a commutative diagram linking twisted correspondences, atlases, and Cartan subalgebras, enabling reconstruction in certain cases and providing criteria for when different objects yield isomorphic $C^*$-correspondences. The paper also explores concrete applications and examples, including twisted coverings over spheres, the existence of coverings with nontrivial associated vector bundles, and local versus global reconstruction/rigidity phenomena, thereby connecting noncommutative and fiber-bundle viewpoints in operator algebra theory.

Abstract

We introduce twisted topological correspondences, which generalize both Katsura's topological correspondences as well as the twisted topological graphs introduced by Li. We show that, up to isomorphism, they are in bijection with certain principal bundles. This makes it possible to study topological correspondences using the machinery of principal and fiber bundles. We show how to associate a $C^*$-correspondence to a twisted topological correspondence, and give two different characterizations of the $C^*$-correspondences arising that way. The first one is the existence of an atlas of the vector bundle associated to the $C^*$-correspondence whose transition functions take values in a certain subgroup of the unitary group $U(n)$, and which is in some sense compatible with the left action. The other characterization is in terms of Cartan subalgebras in the compact operators on the $C^*$-correspondence. We use our findings to prove rigidity results of the $C^*$-correspondences associated to twisted topological correspondences.

Twisted topological correspondences and Cartan subalgebras

TL;DR

This work introduces twisted topological correspondences as a unifying geometric framework for objects generalizing Katsura’s topological correspondences and Li’s twisted graphs. It proves a bijection between twisted correspondences and principal bundles valued in unitary normalizers, and shows how these structures yield -correspondences with two equivalent characterizations: (i) atlases with -valued transitions compatible with the left action, and (ii) Cartan subalgebras in the compacts that contain the left action; these viewpoints lead to rigidity results. A central outcome is a commutative diagram linking twisted correspondences, atlases, and Cartan subalgebras, enabling reconstruction in certain cases and providing criteria for when different objects yield isomorphic -correspondences. The paper also explores concrete applications and examples, including twisted coverings over spheres, the existence of coverings with nontrivial associated vector bundles, and local versus global reconstruction/rigidity phenomena, thereby connecting noncommutative and fiber-bundle viewpoints in operator algebra theory.

Abstract

We introduce twisted topological correspondences, which generalize both Katsura's topological correspondences as well as the twisted topological graphs introduced by Li. We show that, up to isomorphism, they are in bijection with certain principal bundles. This makes it possible to study topological correspondences using the machinery of principal and fiber bundles. We show how to associate a -correspondence to a twisted topological correspondence, and give two different characterizations of the -correspondences arising that way. The first one is the existence of an atlas of the vector bundle associated to the -correspondence whose transition functions take values in a certain subgroup of the unitary group , and which is in some sense compatible with the left action. The other characterization is in terms of Cartan subalgebras in the compact operators on the -correspondence. We use our findings to prove rigidity results of the -correspondences associated to twisted topological correspondences.

Paper Structure

This paper contains 23 sections, 35 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Summary of the paper
  3. Twisted covering maps and their associated principal bundles
  4. Principal bundles and fiber bundles
  5. Constructing morphisms of principal bundles
  6. Twisted covering maps
  7. Principal $\mathbb{T}^n\rtimes S_n$-bundles and twisted coverings
  8. Isomorphism of Hilbert modules
  9. Correspondences
  10. Obtaining an atlas from a correspondence
  11. Obtaining a topological correspondence from an atlas
  12. Equivalence of atlases
  13. Cartan subalgebras in the compacts
  14. Cartan subalgebras and atlases
  15. Cartan subalgebras and correspondences
  16. ...and 8 more sections

Key Result

Lemma 2.1

The map $\rho$ induces a principal bundle morphism from $\mathcal{P}$ to $\mathcal{Q}$. If $\rho$ is surjective, then this morphism is as well.

Theorems & Definitions (100)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 90 more