Positivity 11 (2007), 617–625
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/040617-9, published online September 26, 2007
Classically Normal Pure States
Charles Akemann and Nik Weaver
Abstract. 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 pM p is a factor of type I
Mathematics Subject Classiﬁcation (2000). 46L10.
Keywords. Kadison-Singer, pure states, classically normal.
Deﬁnition. A pure state f of a von Neumann algebra M is called classically nor-
mal 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 classi-
cally normal if, for every von Neumann subalgebra C of M, either f is not multi-
plicative on C, or else there is a minimal projection q in C such that f(q)=1and
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  suggested that every pure state on B(H) would restrict to a pure
state on some maximal abelian self-adjoint subalgebra (aka MASA). Anderson 
formulated the stronger conjecture that every pure state on B(H)isoftheform
f(a) = lim
for some orthonormal basis (e
) and some ultraﬁlter U over
the natural numbers N. Using the continuum hypothesis, we showed in  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.