Non-commutative disintegrations: Existence and uniqueness in finite dimensions

Arthur J. Parzygnat, Benjamin P. Russo

Type: Article

Publication Date: 2023-07-15

Citations: 1

DOI: https://doi.org/10.4171/jncg/493

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 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^* -algebras, and we give formulas for the associated disintegrations.

Locations

arXiv (Cornell University) - View - PDF
OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information) - View - PDF
Journal of Noncommutative Geometry - View - PDF

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+	C[]-algebras and W[]-algebras	1971	正一郎境
+	Real Analysis: Modern Techniques and Their Applications	1984	Gerald B. Folland
+	Real and Complex Analysis.	1987	G. A. Garreau Walter Rudin
+	Completely Bounded Maps and Operator Algebras	2003	Vern I. Paulsen
+	DISINTEGRATION OF MEASURES AND LIFTING	1973	S. D. Chatterji
+ PDF Chat	Positive Linear Maps of Operator Algebras	2012	Erling Størmer
+ PDF Chat	Inner product modules over 𝐵*-algebras	1973	William L. Paschke
+	Completely positive linear maps on complex matrices	1975	Man-Duen Choi
+	A Generalized Schwarz Inequality and Algebraic Invariants for Operator Algebras	1952	Richard V. Kadison
+	Nonstandard techniques in lifting theory	2011	Jesse E. Miller