Atomic Boolean Subspace Lattices and Applications to the Theory of Bases

This book provides a bridge between the theory of bases of Banach spaces and the study of certain types (reflexive, non-self-adjoint) of operator algebras, offering a viewpoint common to both areas. The authors give a characterization of those families of sub-spaces of a Banach space that arise as the set of atoms of an atomic Boolean subspace lattice (ABSL). They obtain new examples of ABSLs, including some with one-dimensional atoms. The latter are shown to arise precisely from strong M-bases of the underlying space. The authors also discuss, for any given ABSL, the question of the strong-operator density of the sub-algebra of finite-rank operators in the algebra of all operators leaving every atom invariant; some affirmative results are presented. On a separable Hilbert space, the given ABSL satisfies this density property if and only if a certain extremely non-commutative "factor" of it does. In addition, several other areas of investigation in the theory of ABSLs are considered, including "selection" from atoms, "slicing" of atoms, and the double commutant property. The authors also provide many examples.

Quasi-direct sums; Strong rank one density property, Meshed products; Strong M-bases; Selecting and slicing, The double commutant.