States with equivalent supports

ANDRUCHOW, ESTEBAN; VARELA, ALEJANDRO

JOURNAL OF OPERATOR THEORY

Theta

Lugar: Bucarest; Año: 2005 vol. 53 p. 35 - 48

0379-4024

Let B be a von Neumann algebra and X a C* Hilbert B-module. If  p in B is a projection, denote by Sp ={x in X : <x,x> =p}, the p-sphere of X. For f a state of B with support p  in B and x in Sp, consider the state f_x of L_B(X) given by f_x(t)= (<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 in R_0, regarded with the norm topology (of the conjugate space of R_0), has trivial homotopy groups of all orders.