Homotopy of state orbits

1999/11/17 by E. Andruchow, Andruchow, E., A. Varela +1
Mathematics · #46L30 #58B05 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:46L30 #msc:58B05

paper · pdf · doi:10.48550/arxiv.math/9911124

12 pages, Latex

arxiv created 1999/11/17 · arxiv updated 2009/11/30

Abstract

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 → 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 ∘ Ad(u^*): u∈ 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).

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