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Continuity of the support of a state

Author: Esteban Andruchow
Journal: Proc. Amer. Math. Soc. 130 (2002), 3565-3570
MSC (2000): Primary 46L30, 46L05, 46L10
Published electronically: July 2, 2002
MathSciNet review: 1920035
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Abstract: Let ${\mathcal A}$ be a finite von Neumann algebra and $p\in {\mathcal A}$ a projection. It is well known that the map which assigns its support projection to a positive normal functional of ${\mathcal A}$ is not continuous. In this note it is shown that if one restricts to the set of positive normal functionals with support equivalent to a fixed $p$, endowed with the norm topology, and the set of projections of ${\mathcal A}$ is considered with the strong operator topology, then the support map is continuous. Moreover, it is shown that the support map defines a homotopy equivalence between these spaces. This fact together with previous work implies that, for example, the set of projections of the hyperfinite II$_1$ factor, in the strong operator topology, has trivial homotopy groups of all orders $n\ge 1$.

References [Enhancements On Off] (What's this?)

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Additional Information

Esteban Andruchow
Affiliation: Instituto de Ciencias, Universidad Nacional de Gral. Sarmiento, J. M. Gutierrez entre J.L. Suarez y Verdi, (1613) Los Polvorines, Argentina

Keywords: State space, support projection
Received by editor(s): September 1, 2000
Published electronically: July 2, 2002
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society