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of a C*-algebra
is called irredundant if no
belongs to the C*-subalgebra of
. 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.
(an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of
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
of cardinality continuum contains an irredundant subcollection of cardinality continuum.
Other partial results and more open problems are presented.
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