c © Copyright by Theta, 48 HOMOTOPY OF STATE ORBITS

Abstract. Let M be a von Neumann algebra, ϕ a faithful normal state and denote by Mϕ the fixed point algebra of the modular group of ϕ. Let UM and UMϕ be the unitary groups of M and M ϕ. In this paper we study the quotient Uϕ = UM/UMϕ endowed with two natural topologies: the one induced by the usual norm of M (called here usual topology of Uϕ), and the one induced by the pre-Hilbert C∗-module norm given by the ϕ-invariant conditional expectation Eϕ: M → Mϕ (called the modular topology). It is shown that Uϕ 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 Eϕ is finite. The set Uϕ can be regarded as a model for the unitary orbit {ϕ ◦ Ad(u∗) : u ∈ UM} of ϕ, and either with the usual or the modular it can be embedded continuously in the conjugate space M ∗ (although not as a topological submanifold)