Classically Normal Pure States

A pure state f of a von Neumann algebra M is called classically normal if f is normal on any von Neumann subalgebra of M on which f is multiplicative. Assuming the continuum hypothesis, a separably represented von Neumann algebra M has classically normal, singular pure states iff there is a central projection p ∈ M such that pMp is a factor of type I∞, II, or III. DEFINITION: A pure state f of a von Neumann algebra M is called classically normal if f is normal on any von Neumann subalgebra of M (”subalgebra” implies the same unit) on which f is multiplicative. By Lemma 0.2 below, a pure state f on a von Neumann algebra M is classically normal if, for every von Neumann subalgebra C of M, either f is not multiplicative on C, or else there is a minimal projection q in C such that f(q) = 1 and q is central in C. Using the continuum hypothesis and a transfinite construction, in Theorem 0.7 we show the existence of classically normal, singular pure states on all infinite dimensional factors acting on a separable Hilbert space. Corollary 0.8 contains the easy ”only if” part of the main result. Here is some history. Let H be a separable infinite-dimensional Hilbert space and let B(H) denote the algebra of all bounded linear operators on H . Kadison and Singer [12] suggested that every pure state on B(H) would restrict to a pure state on some maximal abelian self-adjoint subalgebra (aka MASA). Anderson [9] formulated the stronger conjecture that every pure state on B(H) is of the form f(a) = limU〈aen, en〉 for some orthonormal basis (en) and some ultrafilter U over the natural numbers N. Using the continuum hypothesis, we showed in [6] that these conjectures are false by showing that there is a pure state f on B(H) that is not multiplicative on any MASA. The argument in the key lemma of that paper used powerful results about the Calkin algebra, so finding the ”right” definitions and proofs for general von Neumann algebras took some time. Our construction of a classically normal pure state will be by transfinite induction, just as in [6]. The difference will be in the proofs of the Lemmas that allow the transfinite construction to go through. We start with some easy facts. Lemma 0.1. Let f denote a state on a C*-algebra B in which the linear combinations of the projections are dense. f is multiplicative on B iff f(p) ∈ {0, 1} for all projections p ∈ M. Proof. Suppose that a, b ∈ B and f(ab) 6= f(ba) WLOG we can assume that a = ∑ sjpj , b = ∑ tiqi, finite linear combinations of projections since the map (a, b) → (ab − ba) → f(ab − ba) is continuous, so we only need to show that it annihilates a dense set in A×A. Then f(ba) = ∑ sjtjf(qipj).