DC FieldValueLanguage
dc.contributor.authorCalderón, Danielen_US
dc.contributor.authorFarah, Ilijasen_US
dc.date.accessioned2023-06-27T09:38:56Z-
dc.date.available2023-06-27T09:38:56Z-
dc.date.issued2023-
dc.identifier.issn0022-1236-
dc.identifier.urihttp://researchrepository.mi.sanu.ac.rs/handle/123456789/5103-
dc.description.abstractBy 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.en_US
dc.publisherElsevieren_US
dc.relation.ispartofJournal of Functional Analysisen_US
dc.subjectForcing | Jensen's diamond | Naimark's problem | Representations of C⁎-algebrasen_US
dc.titleCan you take Akemann–Weaver's ⋄ℵ1 away?en_US
dc.typeArticleen_US
dc.identifier.doi10.1016/j.jfa.2023.110017-
dc.identifier.scopus2-s2.0-85161058012-
dc.contributor.affiliationMathematical Institute of the Serbian Academy of Sciences and Artsen_US
dc.relation.firstpage110017-
dc.relation.issue5-
dc.relation.volume285-
dc.description.rank~M21-
item.cerifentitytypePublications-
item.openairetypeArticle-
item.fulltextNo Fulltext-
item.grantfulltextnone-
item.openairecristypehttp://purl.org/coar/resource_type/c_18cf-
crisitem.author.orcid0000-0001-7703-6931-
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