Non-commutative disintegrations: Existence and uniqueness in finite dimensions
Arthur J. Parzygnat
Institut des Hautes Études Scientifiques, Bures-sur-Yvette, FranceBenjamin P. Russo
Oak Ridge National Laboratory, USA
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Abstract
Motivated by advances in categorical probability, we introduce non-commutative almost everywhere (a.e.) equivalence and disintegrations in the setting of -algebras. We show that -algebras (resp. -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 -algebras, and we give formulas for the associated disintegrations.
Cite this article
Arthur J. Parzygnat, Benjamin P. Russo, Non-commutative disintegrations: Existence and uniqueness in finite dimensions. J. Noncommut. Geom. 17 (2023), no. 3, pp. 899–955
DOI 10.4171/JNCG/493