Hilbert Spaces Without Countable AC

Abstract

This article examines Hilbert spaces constructed from sets whose existence is incompatible with the Countable Axiom of Choice (CC). Our point of view is twofold: (1) We examine what can and cannot be said about Hilbert spaces and operators on them in ZF set theory without any assumptions of Choice axioms, even the CC. (2) We view Hilbert spaces as ``quantized'' sets and obtain some set-theoretic results from associated Hilbert spaces.

52 pages. The sections of the paper that deal with C*-algebras without countable AC will appear in a separate paper, with additional results. Final version to appear in the Münster Journal of Mathematics