Connections on statistical manifolds of density operators by geometry of noncommutative L-p-spaces

Let N be a statistical manifold of density operators, with respect to an n.s.f. trace tau on a semifinite von Neumann algebra M. If S-p is the unit sphere of the noncommutative space L-p(M, tau), using the noncommutative Amari embedding rho is an element of N --> rho(1/p) is an element of S-p, we define a noncommutative alpha-bundle-connection pair (F-alpha,del(alpha)), by the pullback technique. In the commutative case we show that it coincides with the construction of nonparametric Amari-Centsov alpha-connection made in Ref. 8 by Gibilisco and Pistone