Authors: Calderón, Daniel
Farah, Ilijas 
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Can you take Akemann–Weaver's ⋄ℵ1 away?
Journal: Journal of Functional Analysis
Volume: 285
Issue: 5
First page: 110017
Issue Date: 2023
Rank: ~M21
ISSN: 0022-1236
DOI: 10.1016/j.jfa.2023.110017
By Glimm's dichotomy, a separable, simple C⁎-algebra has continuum many unitarily inequivalent irreducible representations if, and only if, it is non-type I while all of its irreducible representations are unitarily equivalent if, and only if, it is type I. Naimark asked whether the latter equivalence holds for all C⁎-algebras. In 2004, Akemann and Weaver gave a negative answer to Naimark's problem using Jensen's Diamond Principle ⋄ℵ1, a powerful diagonalization principle that implies the Continuum Hypothesis (CH). By a result of Rosenberg, a separably represented, simple C⁎-algebra with a unique irreducible representation is necessarily of type I. We show that this result is sharp by constructing an example of a separably represented, simple C⁎-algebra that has exactly two inequivalent irreducible representations, and therefore does not satisfy the conclusion of Glimm's dichotomy. Our construction uses a weakening of Jensen's ⋄ℵ1, denoted ⋄Cohen, that holds in the original Cohen's model for the negation of CH. We also prove that ⋄Cohen suffices to give a negative answer to Naimark's problem. Our main technical tool is a forcing notion that generically adds an automorphism of a given C⁎-algebra with a prescribed action on its space of pure states.
Keywords: Forcing | Jensen's diamond | Naimark's problem | Representations of C⁎-algebras
Publisher: Elsevier

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