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Continuity Of The Support Of A State

By Esteban Andruchow Instituto and Esteban Andruchow


Let A be a finite von Neumann algebra and p 2 A a projection. It is well known that the map which assigns to a positive normal functional of A its support projection, 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 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 imply 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 1. Keywords: State space, support projection. 1 Introduction It is well known that the function which assigns to each positive normal functional ' in a von Neumann algebra A, its support projection supp('), is not continuous, in any topology ot..

Topics: State space, support projection
Year: 2007
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