Non-commutative disintegrations: Existence and uniqueness in finite dimensions

Abstract

Motivated by advances in categorical probability, we introduce non-commutative almost everywhere (a.e.) equivalence and disintegrations in the setting of $C^{\ast }$-algebras. We show that $C^{\ast }$-algebras (resp. $W^{\ast }$-algebras) and a.e. equivalence classes of 2-positive (resp. positive) unital maps form a category. We prove that 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 $C^{\ast }$-algebras, and we give formulas for the associated disintegrations.