C*- Modular Vector States

Abstract

Let B be a C^*-algebra and X a Hilbert C^*B-module. If p ∈ B is a projection, let S_p (X) = {x∈ X : (x,x) =p} be the p-sphere of X. For φ a state of B with support p in B and x ∈ S_p(X), consider the modular vector state φ_x of L_B(X) given by φ _x(t)=φ ((x,t(x))). The spheres S_p(X) provide fibrations S_p (X)→ Ο_φ = {φ_x: x ∈ S_p(X)}, x→φ_x, and S_p(X) x {states with support } p}→Σ_{p,x}={ modular vector states}, (x, φ)→φ_x. These fibrations enable us to examine the homotopy type of the sets of modular vector states, and relate it to the homotopy type of unitary groups and spaces of projections. We regard modular vector states as generalizations of pure states to the context of Hilbert C*-modules, and the above fibrations as generalizations of the projective fibration of a Hilbert space.