B(H) has a pure state that is not multiplicative on any masa.

Akemann, Charles; Weaver, Nik · Proc Natl Acad Sci U S A · 2008

Where this comes from

Abstract

Assuming the continuum hypothesis, we prove that Bernoulli function(H) has a pure state whose restriction to any masa is not pure. This resolves negatively old conjectures of Kadison and Singer and of Anderson.