On Bures distance and *-algebraic transition probability between inner derived positive linear forms over W*-algebras

On a W*-algebra M, for given two positive linear forms nu, rho is an element of M+* and algebra elements a, b is an element of M, a variational expression for the Bures distance d(B)(nu(a), rho(b)) between the inner derived positive linear forms nu(a)=nu(a*.a) and rho(b)=rho(b*.b) is obtained. Along with the proof of the formula, also an earlier result of S. Gudder on noncommutative probability will be slighly extended. Also, the given expression of the Bures distance relates nicely to the system of seminorms proposed by D. Buchholz which occurs, along with the problem of estimating the so-called 'weak intertwiners', in algebraic quantum field theory. In the last section, some optimization problem will be considered.