Square-cube law: Difference between revisions

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'''[[Mark Naimark|Naimark]]'s Problem''' is a question in [[functional analysis]]. It asks whether every [[C*-algebra]] that has only one [[irreducible representation|irreducible <math> * </math>-representation]] up to [[Self-adjoint operator#Spectral theorem|unitary equivalence]] is [[isomorphic]] to the <math> * </math>-algebra of [[Compact operator on Hilbert space|compact operator]]s on some (not necessarily separable) [[Hilbert space]].
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The problem has been solved in the affirmative for special cases (specifically for separable and Type-I C*-algebras). {{harvtxt|Akemann|Weaver|2004}} used the [[diamond principle|<math> \diamondsuit </math>-Principle]] to construct a [[C*-algebra]] with <math> \aleph_{1} </math> generators that serves as a counterexample to Naimark's Problem. More precisely, they showed that the statement "'''There exists a counterexample to Naimark's Problem that is generated by [[Aleph One|<math> \aleph_{1} </math>]] elements'''" is <u>independent</u> of the axioms of [[Zermelo-Fraenkel set theory|Zermelo-Fraenkel Set Theory]] and the [[axiom of choice|Axiom of Choice]] (<math> \mathsf{ZFC} </math>).
 
==See also==
*[[List of statements undecidable in ZFC|List of statements undecidable in <math> \mathsf{ZFC} </math>]]
*[[Gelfand–Naimark theorem|Gelfand-Naimark Theorem]]
 
==External links==
*{{Citation | last1 = Akemann | first1 = Charles | last2 = Weaver | first2 = Nik | title = Consistency of a counterexample to Naimark's problem | doi = 10.1073/pnas.0401489101 | mr = 2057719 | year = 2004 | journal = [[Proceedings of the National Academy of Sciences|Proceedings of the National Academy of Sciences of the United States of America]] | volume = 101 | issue = 20 | pages = 7522–7525 | arxiv = math.OA/0312135}}
 
[[Category:Conjectures]]
[[Category:C*-algebras]]
[[Category:Independence results]]
 
 
{{Mathanalysis-stub}}

Latest revision as of 16:02, 8 January 2015

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