Inductive Limits of Noncommutative Cartan Inclusions

TL;DR

This paper establishes conditions under which inductive limits of noncommutative Cartan inclusions retain their structure, extending previous results and simplifying proofs in the theory of operator algebras.

Contribution

It proves that inductive limits of aperiodic noncommutative Cartan inclusions are themselves Cartan under certain conditions, and generalizes existing theorems without using twisted étale groupoids.

Findings

Inductive limits preserve noncommutative Cartan structure under injective, normaliser-preserving maps.

Additional conditions yield aperiodic inclusions in the limit, including cases with separable, simple, or Type I Cartan subalgebras.

Provides a new proof of Xin Li's theorem without twisted étale groupoids.

Abstract

We prove that an inductive limit of aperiodic noncommutative Cartan inclusions is a noncommutative Cartan inclusion whenever the connecting maps are injective, preserve normalisers and entwine conditional expectations. We show that under the additional assumption that the inductive limit Cartan subalgebra is either essentially separable, essentially simple or essentially of Type I we get an aperiodic inclusion in the limit. Consequently, we subsume the case where the building block Cartan subalgebras are commutative and provide a proof of a theorem of Xin Li without passing to twisted \'etale groupoids.