Nonseparable UHF algebras I: Dixmier's problem

TL;DR

This paper investigates whether the three classical definitions of UHF C*-algebras are equivalent in the nonseparable case, providing a complete answer with specific counterexamples and positive results within ZFC set theory.

Contribution

It establishes the conditions under which the definitions are equivalent or not in nonseparable UHF algebras, resolving Dixmier's longstanding problem.

Findings

Two definitions are equivalent in small cardinality cases.

Counterexamples show non-equivalence in other cases.

Results are obtained within standard set theory (ZFC).

Abstract

There are three natural ways to define UHF (uniformly hyperfinite) C*-algebras, and all three definitions are equivalent for separable algebras. In 1967 Dixmier asked whether the three definitions remain equivalent for not necessarily separable algebras. We give a complete answer to this question. More precisely, we show that in small cardinality two definitions remain equivalent, and give counterexamples in other cases. Our results do not use any additional set-theoretic axioms beyond the usual axioms, namely ZFC.