Multi-states on $C^ *$-algebras

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.