Quantum measurable cardinals

Abstract

We investigate states on von Neumann algebras which are not normal but enjoy various forms of infinite additivity, and show that these exist on $B(H)$ if and only if the cardinality of an orthonormal basis of $H$ satisfies various large cardinal conditions. For instance, there is a singular countably additive pure state on $B(l^2(\kappa))$ if and only if $\kappa$ is Ulam measurable, and there is a singular ${<}\,\kappa$-additive pure state on $B(l^2(\kappa))$ if and only if $\kappa$ is measurable. The proofs make use of Farah and Weaver's theory of quantum filters. Applications to Ueda's peak set theorem for von Neumann algebras are discussed in the final section.