Separable C*-algebras Without the Countable Axiom of Choice
Bruce Blackadar, Ilijas Farah
TL;DR
This work investigates the theory of separable C*-algebras in ZF without the Axiom of Choice, developing AC-free proofs of foundational results such as the Gelfand–Naimark theorems, spectra, and continuous functional calculus for commuting normal elements. A central theme is the role of Shoenfield absoluteness, standard Borel spaces, and the projective hierarchy in transferring ZFC-derived facts to ZF for low-complexity statements, along with a detailed examination of states, representations, and pathologies that arise without Choice (e.g., Russell-set constructions). The authors prove that separable C*-algebras have faithful representations on separable Hilbert spaces, and that commutative separable algebras are isomorphic to $C(X)$ for appropriate compact metrizable spaces, while also constructing an AC-free counterexample under a Russell set. They further develop an absoluteness framework to analyze which set-theoretic results about operator algebras remain valid in ZF, and discuss how separability and sigma-completeness of the separable-subalgebra system behave under various choice principles. Overall, the paper clarifies the foundational dependencies of C*-algebra theory on Choice and provides tools for analysts to work effectively in ZF.
Abstract
The goal of this paper is twofold. In addition to the results stated in the next paragraph, we present some classical results on absoluteness relevant to functional analysis that are well known to logicians but not nearly as well advertised as they should be. We show that the theory of separable C*-algebras can be developed in ZF (that is, without using any Choice). This includes proving the Gelfand-Naimark representation theorems as well as the Spectral Mapping Theorem for polynomials and developing continuous functional calculus for commuting normal elements. Some of our proofs are modifications of the standard ones, obtained by avoiding the use of Choice. Some other proofs require new ideas in order to avoid the use of Choice. Yet another batch of proofs proceeds by using the set-theoretic Shoenfield Absoluteness Theorem. This result (well known to logicians but regrettably not as well advertised as it deserves) implies that statements about standard Borel spaces of low quantifier complexity that are provable in ZFC, or even ZFC together with the Continuum Hypothesis are provable in ZF. One of the main objectives of this paper is to present these results in a convenient form that can be utilized by analysts not familiar with set theory. We also show that in the absence of Choice (more precisely, assuming the existence of a Russell set) there is a concretely representable unital commutative \cstar-algebra that is not isomorphic to C(X) for any compact Hausdorff space X. Finally, from the model-theoretic point of view, while the property of having a tracial state is provably axiomatizable in ZFC, it is not provably axiomatizable in ZF+DC.