Journal of Operator Theory

Volume 53, Issue 1, Winter 2005 pp. 35-48.

States with equivalent supports

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

Summary: Let $\b$ be a von Neumann algebra and $X$ a $C^*$ Hilbert $\b$-module. If $p\in \b$ is a projection, denote by $\sp =\{x\in X : \langle x,x\rangle =p\}$, the $p$-sphere of $X$. For $\f$ a state of $\b$ with support $p$ in $\b$ and $x\in \sp$, consider the state $\f_x$ of $\l$ given by $\f_x(t)=\f(\langle x,t(x)\rangle )$. In this paper we study certain sets associated to these states, and examine their topologic properties. As an application of these techniques, we prove that the space of states of the hyperfinite II$_1$ factor ${\mathcal R}_0$, with support equivalent to a given projection $p\in {\mathcal R}_0$, regarded with the norm topology (of the conjugate space of ${\mathcal R}_0$), has trivial homotopy groups of all orders.

Keywords: State space, $C^*$-module.

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