  Let M be a von Neumann algebra, f a faithful normal state and denote by M^f
the fixed point algebra of the modular group of f. Let U_M and U_{M^f} be the
unitary groups of M and M^f. In this paper we study the quotient U_M/U_{M^f}
endowed with two natural topologies: the one induced by the usual norm of M
(called here usual topology), and the one induced by the pre-Hilbert C*-module
norm given by the f-invariant conditional expectation E_f:M \to M^f (called the
modular topology). It is shown that U_M/U_{M^f} is simply connected with the
usual topology. Both topologies are compared, and it is shown that they
coincide if and only if the Jones index of E_f is finite. The set U_M/U_{M^f}
can be regarded as a model for the unitary orbit {f \circ Ad(u^*): u\in U_M} of
f, and either with the usual or the modular it can be embedded continuously in
the conjugate space M* (although not as a topological submanifold).
