  In case of a standard form vN-algebra, 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
extend 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.
