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Homotopy of state orbits

E. Andruchow·A. Varela
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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 \to 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 \circ Ad(u^*): u\in 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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E. Andruchow, A. Varela. 1999-11-17. Homotopy of state orbits. https://arxiv.org/abs/math/9911124

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