Quantum measurable cardinals

Abstract

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 (κ)) if and only if κ is Ulam measurable, and there is a singular < κ-additive pure state on B(l 2 (κ)) if and only if κ is measurable. The proofs make use of Farah and Weaver's theory of quantum filters [10]. We can generalize some of these characterizations to arbitrary von Neumann algebras. Applications to Ueda's peak set theorem for von Neumann algebras are discussed in the final section.

Ueda’s peak set theorem for general von Neumann algebras

“…The existence of regular singular states or such regular measures is generally believed to be consistent with ZFC set theory. Indeed as explained in [13] it is believed to be consistent with ZFC set theory that the latter 'first cardinal' is ≤ the cardinality of the real numbers. On the other hand, since any cardinal on which there is a singular probability measure dominates the 'first cardinal' above, it follows that if M ⊂ B(H) where dim(H) is smaller than any real-valued measurable cardinal (or if measurable cardinals do not exist), then Ueda's peak set theorem holds for M (and taking A = M ).…”

“…Indeed certain cases of Ueda's peak set theorem, for a von Neumann algebra M , may be seen as 'set theoretic statements' about M that require the sets to not be 'too large'. These issues are discussed in Section 6, and this also led to a sequel paper with Nik Weaver [13]. Some of the ramifications of [13] are described at the end of the present paper, for example that that work indicates that one cannot generalize Ueda's peak set theorem in ZFC much beyond the σ-finite case (not even to l ∞ (R)).…”

“…These issues are discussed in Section 6, and this also led to a sequel paper with Nik Weaver [13]. Some of the ramifications of [13] are described at the end of the present paper, for example that that work indicates that one cannot generalize Ueda's peak set theorem in ZFC much beyond the σ-finite case (not even to l ∞ (R)). Thus the main result of our paper is somewhat sharp.…”

“…The following result, which characterizes peak projections in subalgebras of von Neumann algebras, will also be used in [13].…”

“…Acknowledgments. We are grateful to Nik Weaver for a very helpful and lengthy conversation shortly before distribution of an earlier version of the present paper, which led to [13], a paper largely devoted to quantum measure theory in the sense of [23], quantum cardinals, and various continuity properties of states on von Neumann algebras. In Section 9 of that paper the set theoretic issues associated with Ueda's peak set theorem for von Neumann algebras are explored more fully, as mentioned above.…”