DC Field | Value | Language |
---|---|---|
dc.contributor.author | Calderón, Daniel | en_US |
dc.contributor.author | Farah, Ilijas | en_US |
dc.date.accessioned | 2023-06-27T09:38:56Z | - |
dc.date.available | 2023-06-27T09:38:56Z | - |
dc.date.issued | 2023 | - |
dc.identifier.issn | 0022-1236 | - |
dc.identifier.uri | http://researchrepository.mi.sanu.ac.rs/handle/123456789/5103 | - |
dc.description.abstract | 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. | en_US |
dc.publisher | Elsevier | en_US |
dc.relation.ispartof | Journal of Functional Analysis | en_US |
dc.subject | Forcing | Jensen's diamond | Naimark's problem | Representations of C⁎-algebras | en_US |
dc.title | Can you take Akemann–Weaver's ⋄ℵ1 away? | en_US |
dc.type | Article | en_US |
dc.identifier.doi | 10.1016/j.jfa.2023.110017 | - |
dc.identifier.scopus | 2-s2.0-85161058012 | - |
dc.contributor.affiliation | Mathematical Institute of the Serbian Academy of Sciences and Arts | en_US |
dc.relation.firstpage | 110017 | - |
dc.relation.issue | 5 | - |
dc.relation.volume | 285 | - |
dc.description.rank | ~M21 | - |
item.grantfulltext | none | - |
item.cerifentitytype | Publications | - |
item.fulltext | No Fulltext | - |
item.openairecristype | http://purl.org/coar/resource_type/c_18cf | - |
item.openairetype | Article | - |
crisitem.author.orcid | 0000-0001-7703-6931 | - |
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