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Large Irredundant Sets in Operator Algebras

  • Clayton Suguio Hida (a1) and Piotr Koszmider (a2)

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.

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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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Large Irredundant Sets in Operator Algebras

  • Clayton Suguio Hida (a1) and Piotr Koszmider (a2)

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