A study on the geometry of pairs of positive linear forms, algebraic transition probability and geometrical phase over non-commutative operator algebras. I
type: Item
instance of: scholarly article
title: A study on the geometry of pairs of positive linear forms, algebraic transition probability and geometrical phase over non-commutative operator algebras. I
zbMATH Open document ID: 0792.46037
DOI: 10.4171/ZAA/604
author: Peter M. Alberti
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1993-10-13T00:00:00Z
review text: 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.
reviewed by: George A. Elliott
Mathematics Subject Classification ID: 46L05
Mathematics Subject Classification ID: 46L10
Mathematics Subject Classification ID: 46L51
Mathematics Subject Classification ID: 46L53
Mathematics Subject Classification ID: 46L54
zbMATH DE Number: 428727
zbMATH Keywords: global phase
zbMATH Keywords: geometrical phase
zbMATH Keywords: phase group
zbMATH Keywords: holonomy group of a normal state
zbMATH Keywords: geometry of pairs of positive linear forms on unital \(C^*\)- algebras
zbMATH Keywords: vector representatives
zbMATH Keywords: Bures distance
zbMATH Keywords: algebraic transition probability
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 428727