Large irredundant sets in operator algebras

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 $2^\omega=\omega_1$ 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.