A note on the kadison-singer problem

Abstract

Let H be a separable Hilbert space with a fixed orthonormal basis (e(n)) n >= 1 and B(H) be the full von Neumann algebra of the bounded linear operators T : H -> H. identifying l(infinity) = C(beta N) with the diagonal operators, we consider C(beta N) as a subalgebra of B(H). For each t is an element of beta N, let [delta(t)] be the set of the states of B(H) that extend the Dirac measure delta(t). Our main result shows that, for each t in beta N, the set [delta(t)] either lies in a finite dimensional subspace of B(H)* or else it must contain a homeomorphic copy of beta N.