Continuity of the Support of a State

Let A be a nite 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 xed 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 deenes a homotopy equivalence between these spaces. This fact together with previous work imply that, for example, the set of projections of the hyperrnite II 1 factor, in the strong operator topology, has trivial homotopy groups of all orders n 1.