Plymen, Roger J.
(1968)
C*-algebras and Mackey's axioms.
*Communications in Mathematical Physics*, 8 (2), 132-146.
(doi:10.1007/BF01645801).

## Abstract

A non-commutative version of probability theory is outlined, based on the concept of a ?*-algebra of operators (sequentially weakly closed C*-algebra of operators). Using the theory of ?*-algebras, we relate the C*-algebra approach to quantum mechanics as developed by Kadison with the probabilistic approach to quantum mechanics as axiomatized by Mackey. The ?*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by the W*-algebra approach. By considering the ?*-algebra, rather than the von Neumann algebra, generated by the given C*-algebra A in its reduced atomic representation, we show that a difficulty encountered by Guenin concerning the domain of a state can be resolved.

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