CONTINUITY OF THE SUPPORT OF A STATE * Esteban Andruchow Instituto de Ciencias – Univ. Nac. de Gral. Sarmiento Argentina Abstract Let A be a finite von Neumann algebra and p ∈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 func- tional ϕ in a von Neumann algebra A, its support projection supp(ϕ), is not continuous, in any topology other than the coarse topology. This is the case even if A is finite dimensional. In fact, for any von Neumann algebra, it holds that the normal and faithful functionals are dense in norm among all the normal positive functionals (see [2]). However, a simple spectral argument shows that for matrix algebras, if one considers only functionals which have a priori equivalent supports, then the support function is continuous. The * 1991 Mathematics Subject Classification: 46L30, 46L05, 46L10. 1