Homotopy of State Orbits 1

Let M be a von Neumann algebra, ' a faithful normal state and denote by M ' the xed point algebra of the modular group of '. Let U M and U M ' be the unitary groups of M and M '. In this paper we study the quotient U ' = U M =U M ' endowed with two natural topologies: the one induced by the usual norm of M (called here usual topology of U '), and the one induced by the pre-Hilbert C-module norm given by the '-invariant conditional expectation E ' : M ! M ' (called the modular topology). It is shown that U ' 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 ' is nite. The set U ' can be regarded as a model for the unitary orbit f' Ad(u) : u 2 U M g of ', and either with the usual or the modular it can be embedded continuously in the conjugate space M (although not as a topological submanifold).