Traces, Dispersions of States and Hidden Variables

ISSN： 0894-9875

Source： Foundations of Physics Letters, Vol.17, Iss.6, 2004-11, pp. : 581-597

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Abstract

The interplay between the tracial property and minimality of dispersions of states on projections of von Neumann algebras and C^*-algebras is investigated. Let &phis; be a state on a C^*-algebra A with the projection structure P(A). The dispersion σ(&phis;) is defined as σ(&phis;) = sup{&phis;(P) − &phis;(P)² | P ɛ P(A)}. It is proved that σ(&phis;) ≥ 2/9 whenever &phis; is a state on a real rank zero C^*-algebra with no nonzero abelian representation. New characterization of traces in terms of dispersions is proved: A state on a von Neumann algebra without abelian and Type I₂ direct summands is a trace if and only if &phis; has the minimal dispersion on all 3x3 matrix substructures. A similar characterization of semifinite normal traces on von Neumann algebras is obtained. The connection between unitary invariance of states and minimal dispersion property on C^*-algebras is studied. Besides providing a new characterization of trace in terms of physically relevant properties, the existing results on hidden variables in W^*- and C^*-formalism of quantum mechanics are strengthen.