A study on the geometry of pairs of positive linear forms, algebraic transition probability and geometrical phase over non-commutative operator algebras. I
Some aspects of the geometry of pairs of positive linear forms on unital \(C^*\)-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 the first part extend early results of H. Araki and are intimately related to such functors as the Bures distance and the algebraic transition probability considered by A. Uhlmann and others. The results of the first part will be used to discuss and to investigate some extensions of geometrical notions, which recently have been found to be of interest in Mathematical Physics in the context of the problems of the so-called geometrical phase. The concepts of the global phase, the phase group and the holonomy group of a normal state of a vN-algebra will be introduced and discussed.
geometry of pairs of positive linear forms on unital \(C^*\)- algebras
scientific article; zbMATH DE number 428727