Angles Between Intermediate Operator Subalgebras

TL;DR

This paper introduces and studies the concept of interior and exterior angles between compatible intermediate subalgebras in W*- and C*-algebra inclusions, providing new tools for analyzing their geometric relations.

Contribution

It defines interior and exterior angles for intermediate subalgebras in W*- and C*-algebras, extending previous notions and demonstrating stability under tensor products.

Findings

Defined interior and exterior angles for W*-subalgebras

Extended the concept to C*-algebras with finite Watatani index

Proved stability of the interior angle under minimal tensor products

Abstract

Motivated by [2] and [5], the notions of interior and exterior angles between a pair of compatible intermediate W*-subalgebras of an inclusion of W*-algebras with a normal conditional expectation with finite probabilistic index are introduced. This is then employed effectively to define the interior angle between a pair of compatible intermediate C*-subalgebras of an inclusion of non-unital C*-algebras with a conditional expectation with finite Watatani index. It is also shown that the interior angle is stable under the minimal tensor product of unital C*-algebras.