Abstract
A non-commutative version of probability theory is outlined, based on the concept of aΣ*-algebra of operators (sequentially weakly closedC*-algebra of operators). Using the theory ofΣ*-algebras, we relate theC*-algebra approach to quantum mechanics as developed byKadison with the probabilistic approach to quantum mechanics as axiomatized byMackey. TheΣ*-algebra approach to quantum mechanics includes the case of classical statistical mechanics; this important case is excluded by theW*-algebra approach. By considering theΣ*-algebra, rather than the von Neumann algebra, generated by the givenC*-algebraA in its reduced atomic representation, we show that a difficulty encountered byGuenin concerning the domain of a state can be resolved.
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Plymen, R.J. C*-algebras and Mackey's axioms. Commun.Math. Phys. 8, 132–146 (1968). https://doi.org/10.1007/BF01645801
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DOI: https://doi.org/10.1007/BF01645801