Homotopy of State Orbits

Let M be a von Neumann algebra, φ a faithful normal state and denote by M the fixed point algebra of the modular group of φ. Let UM and UMφ be the unitary groups of M and M. 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 → M (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).