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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Multi-states on $ C\sp *$-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 $ {C^ * }$-algebra as a matrix ordered space. It is shown that an $ n \times n$ matrix of linear functionals of a $ {C^ * }$-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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1989-0972233-2
PII: S 0002-9939(1989)0972233-2
Keywords: Matrix order, $ n$-positive linear functional, $ n$-state, representation engendered by an $ n$-positive linear functional, unitary diagonalization
Article copyright: © Copyright 1989 American Mathematical Society