Multi-states on -algebras

Author:
Alexander Kaplan

Journal:
Proc. Amer. Math. Soc. **106** (1989), 437-446

MSC:
Primary 46L30

MathSciNet review:
972233

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Abstract: This paper is concerned with the study of the dual of a -algebra as a matrix ordered space. It is shown that an matrix of linear functionals of a -algebra, satisfying the generalized positivity condition, induces a representation of the algebra that generalizes the classical Gelfand-Naimark-Segal representation. This allows analysis of the relationship between the comparability of cyclic representations of the algebra and the matricial order structure of the dual. We consider the problem of unitary diagonalization of linear functionals and show that positive normal functionals on a matrix algebra over a semifinite von Neumann algebra can always be diagonalized.

**[1]**Richard H. Herman and Masamichi Takesaki,*States and automorphism groups of operator algebras*, Comm. Math. Phys.**19**(1970), 142–160. MR**0270167****[2]**Richard V. Kadison,*Diagonalizing matrices*, Amer. J. Math.**106**(1984), no. 6, 1451–1468. MR**765586**, 10.2307/2374400**[3]**Richard V. Kadison and John R. Ringrose,*Fundamentals of the theory of operator algebras. Vol. I*, Pure and Applied Mathematics, vol. 100, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. Elementary theory. MR**719020**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1989-0972233-2

Keywords:
Matrix order,
-positive linear functional,
-state,
representation engendered by an -positive linear functional,
unitary diagonalization

Article copyright:
© Copyright 1989
American Mathematical Society