A Study on the Geometry of Pairs of Positive linear Forms, Algebraic Transition Probability and Geometrical Phase over Non - Commutative Operator Algebras (I)

Abstract

Some aspects in the geometry of pairs of positive linear forms on unital $C\ast$-algebras are considered. Especially, the geometrical relations among the vector representatives of the forms of such a pair within a representation, where both forms can be realized as vectors simultaneously, are studied and discussed in detail. The results obtained in this part extend early results of H. Araki and are intimately related to such functors like the Bures distance and the algebraic transition probability considered by A. Uhlmann and others. The results will be used to discuss and to investigate sonic extensions of geometrical concepts, which have been found to be of interest recently in Mathematical Physics in context of the problems of the so-called geometrical phase.