Reflexive operator algebras on Banach spaces (Q2510051)

Let \(X\) be a Banach space and let \(B(X)\) be the Banach algebra of all bounded linear operators on \(X\). A subalgebra \(A \subseteq B(X)\) is said to be reflexive if \(\operatorname{alg}\operatorname{Lat} A=A\), where \[ \operatorname{Lat} A=\{F \subseteq X: F \text{ closed and } TF \subseteq F, \text{ for all } T \in A\} \] and \[ \operatorname{alg}\operatorname{Lat} A= \{ T \in B(X): TF \subseteq F, \text{ for all } F \in \operatorname{Lat} A \}. \] Let \(A \subseteq B(X)\) be a unital strongly closed algebra with complemented invariant subspace lattice. In the present paper, the authors show that, if \(A\) contains a complete Boolean algebra of projections of finite uniform multiplicity, and which satisfies the direct sum property, then \(A\) is reflexive. Finally, the authors deduce that such algebras coincide with their bicommutant.