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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A noncommutative probability theory

Authors: S. P. Gudder and R. L. Hudson
Journal: Trans. Amer. Math. Soc. 245 (1978), 1-41
MSC: Primary 46L55; Secondary 46K99, 81B99
MathSciNet review: 511398
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Abstract: A noncommutative probability theory is developed in which no boundedness, finiteness, or ``tracial'' conditions are imposed. The underlying structure of the theory is a ``probability algebra'' $ (\mathcal{a},\omega )$ where $ \mathcal{a}$ is a *-algebra and $ \omega $ is a faithful state on $ \mathcal{a}$. Conditional expectations and coarse-graining are discussed. The bounded and unbounded commutants are considered and commutation theorems are proved. Two classes of probability algebras, which we call closable and symmetric probability algebras are shown to have important regularity properties. The canonical algebra of quantum mechanics is considered in some detail and a strong commutation theorem is proven for this case. Moreover, in this case, isotropic normal states, KMS states, and stable states are defined and characterized.

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Keywords: Noncommutative probability, quantum probability, quantum mechanics, *-algebras, unbounded representations, commutants
Article copyright: © Copyright 1978 American Mathematical Society