States with equivalent supports

Abstract

Let B be a von Neumann algebra and X a C* Hilbert B-module. If p ∈ B is a projection, denote by S_p(X) ={x ∈ X(x,x) =p}, the p-sphere of X. For φ a state of B with support p ∈ B and x ∈ S_p(X), consider the state φ_x of L_B(X) given by φ_x(t)=φ((x,t(x))). In this paper we study certain sets associated to these states, and examine their topologic properties. As an application of these techniques, we prove that the space of states of the hyperfinite II_1 factor R_0, with support equivalent to a given projection p ∈ R_0, regarded with the norm topology (of the conjugate space of R_0), has trivial homotopy groups of all orders.