Non-commutative disintegrations: existence and uniqueness in finite dimensions

TL;DR

This paper develops a framework for non-commutative disintegrations within C*-algebras, establishing existence, uniqueness, and explicit formulas in finite dimensions, advancing categorical probability theory.

Contribution

It introduces non-commutative a.e. equivalence and disintegrations, characterizes their existence in finite-dimensional C*-algebras, and proves their uniqueness.

Findings

Disintegrations are a.e. unique when they exist.

Explicit formulas for disintegrations in finite dimensions.

Characterization of conditions for disintegration existence.

Abstract

Motivated by advances in categorical probability, we introduce non-commutative almost everywhere (a.e.) equivalence and disintegrations in the setting of C*-algebras. We show that C*-algebras (resp. W*-algebras) and a.e. equivalence classes of 2-positive (resp. positive) unital maps form a category. We prove non-commutative disintegrations are a.e. unique whenever they exist. We provide an explicit characterization for when disintegrations exist in the setting of finite-dimensional C*-algebras, and we give formulas for the associated disintegrations.