Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 419-430.

Homotopy of state orbits

Authors: Esteban Andruchow (1), and Alejandro Varela (2)
Author institution: (1) Campus Universitario, Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez y Verdi, (1613) Los Polvorines, Argentina
(2) Campus Universitario, Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J.M. Gutierrez y Verdi, (1613) Los Polvorines, Argentina

Summary: 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 $\uf =U_M/U_{M^\f}$ endowed with two natural topologies: the one induced by the usual norm of $M$ (called here {\it usual topology of} $\uf$), 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 {\it modular topology}). It is shown that $\uf $ 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 $\uf$ 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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