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Large Irredundant Sets in Operator Algebras

Part of: Set theory

Published online by Cambridge University Press:  07 March 2019

Clayton Suguio Hida  and
Piotr Koszmider
Clayton Suguio Hida
Affiliation:
Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, 05314-970, São Paulo, Brazil Email: suguio@ime.usp.br
Piotr Koszmider
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland Email: piotr.koszmider@impan.pl
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Abstract

A subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.

There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.

Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.

Other partial results and more open problems are presented.

Keywords

irredundanceirredundant setscattered C*-algebrathin-tall algebraMcKenzie TheoremOpen Coloring Axiomconstruction scheme

MSC classification

Primary: 03E35: Consistency and independence results
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 4 , August 2020 , pp. 988 - 1023
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

The research of author C. S. H. was partially supported by doctoral scholarships CAPES: 1427540 and CNPq: 167761/2017-0 and 201213/2016-8. The research of the author P. K. was partially supported by grant PVE Ciência sem Fronteiras - CNPq (406239/2013-4).

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