Homotopy of state orbits

ESTEBAN ANDRUCHOW,; ALEJANDRO VARELA

JOURNAL OF OPERATOR THEORY

Theta

Año: 2002 vol. 48 p. 419 - 430

0379-4024

Let M be a von Neumann algebra, f a faithful normal state and denote by M^f thefixed point algebra of the modular group of f. Let UM and UM^f be the unitary groupsof M and M^f. In this paper we study the quotient Uf = UM/UM^f endowed withtwo natural topologies: the one induced by the usual norm of M (called here usualtopology of Uf), and the one induced by the pre-Hilbert C* module norm given bythe f-invariant conditional expectation Ef : M->Mf (called the modular topology).It is shown that Uf is simply connected with the usual topology. Both topologies arecompared, and it is shown that they coincide if and only if the Jones index of Ef isfinite. The set Uf can be regarded as a model for the unitary orbit {f(Ad(u*)) : u is in UM} 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).