Quantum measurable cardinals

TL;DR

This paper explores the existence of special types of states on von Neumann algebras related to large cardinal properties, linking set theory with operator algebras and using quantum filter theory.

Contribution

It establishes a connection between large cardinal conditions and the existence of non-normal, infinitely additive states on $B(H)$, extending the understanding of quantum states in operator algebras.

Findings

Existence of singular countably additive pure states on $B(l^2())$ iff $$ is Ulam measurable.

Existence of singular < $$-additive pure states on $B(l^2())$ iff $$ is measurable.

Application of quantum filter theory to analyze states and implications for Ueda's peak set theorem.

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.