ON BURES DISTANCE OVER STANDARD FORM vN-ALGEBRAS

In case of standard form vN-algebras, the Bures distance is the natural distance between the fibres of implementing vectors at normal positive linear forms. Thereby, it is well-known that to each two normal positive linear forms implementing vectors exist such that the Bures distance is attained by the metric distance of the implementing vectors in question. We discuss to which extent this can remain true if a vector in one of the fibres is considered as fixed. For each nonfinite algebra, classes of counterexamples are given and situations are analyzed where the latter type of result must fail. In the course of the paper, an account of those facts and notions is given, which can be taken as a useful minimum of basic C-algebraic tools needed in order to efficiently develop the fundamentals of Bures geometry over standard form vN-algebras. 1. Basic settings and results 1.1. Definitions and conventions. Throughout the paper, a distance function dB on the positive coneM ∗ + of the bounded linear formsM ∗ over a unital C∗-algebra M will be considered. For normal states on a W∗-algebra dB agrees with the Bures distance function [12]. Therefore, henceforth also dB will be referred to as Bures distance function. We start by defining dB(M |ν, ̺) between ν, ̺ ∈ M∗ +. Definition 1.1. dB(M |ν, ̺) = inf{π,K},φ∈Sπ,M(ν),ψ∈Sπ,M (̺) ‖ψ − φ‖ . Instead of dB(M |ν, ̺) the notation dB(ν, ̺) will often be used. For unital ∗representation {π,K} of M on a Hilbert space {K, 〈·, ·〉} and for μ ∈ M∗ + we let Sπ,M (μ) = {χ ∈ K : μ(·) = 〈π(·)χ, χ〉}. (1.1) In case of Sπ,M (μ) 6= ∅ this set will be referred to as π-fibre of μ. The above infimum extends over all π relative to which both π-fibres exist and, within each such representation, φ and ψ may be varied through all of Sπ,M (ν) and Sπ,M (̺), respectively. The scalar product K × K ∋ {χ, η} 7−→ 〈χ, η〉 ∈ C on the representation Hilbert space by convention is supposed to be linear with respect to the first argument χ, and antilinear in the second argument η, and maps into the complex field C. Let C ∋ z 7→ z̄ be the complex conjugation, and be Rz and |z| the real part and absolute value of z, respectively. The norm of χ ∈ K is given by ‖χ‖ = √ 〈χ, χ〉. For the relating operator and C∗-algebra theory, the reader is referred to the standard monographs, e.g. [16, 28, 21]. For both the C∗-norm of an element x ∈ M as well as for the operator norm of a concrete bounded linear operator x ∈ B(K) the same notation ‖x‖ will be used, 1991 Mathematics Subject Classification. 46L89, 46L10, 58B20. The paper is the completed version of some parts of the lectures on Bures geometry which were held by the first author at ‘Graduiertenkolleg Quantenfeldtheorie’, in spring term 2000. The second author was supported by a ‘Doktorandenförderplatz’ at ITP.