Continuity of the support of a state

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 $n\ge 1$.