Gleason measures on infinite tensor products of Hilbert spaces
Copyright policy )Gleason measures on infinite tensor products of Hilbert spaces
Nowak [Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 22, 393–5 (1974)] has given an example of a consistent (in the sense of Kolmogorov) family of Gleason measures [A. M. Gleason, J. Math. Mech. 6, 885–94 (1957)] {mn} defined over ⊗ni=1Hi which do not extend to a Gleason measure on ⊙∞i=1 ΦHi for a given construction of the infinite tensor product. In this paper we show: (1) In the example of Nowak it is not necessary to assume, as is done, that the Hi are infinite dimensional. (2) That every consistent family developed from pure states, which is the type considered by Nowak, extends over the complete infinite tensor product of von Neumann [Compositio. Math. 6, 1–77 (1938)]. (3) Even if each Hi is two-dimensional and the complete infinite tensor product of von Neumann is used, it is possible to give a simple counterexample to the conjecture that every consistent family of Gleason measures extends by the use of nonpure states.
- Georgia Institute of Technology United States
Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.), General theory of \(C^*\)-algebras, General theory of von Neumann algebras, Inner product spaces and their generalizations, Hilbert spaces, Tensor products in functional analysis
Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.), General theory of \(C^*\)-algebras, General theory of von Neumann algebras, Inner product spaces and their generalizations, Hilbert spaces, Tensor products in functional analysis
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