J-subspace lattices and subspace M-bases
W. E. Longstaff, Oreste Panaia · UWA Profiles and Research Repository (UWA) · 1998
The class of J-lattices was defined in the second author's thesis. A subspace lattice on a Banach space X which is also a J-lattice is called a J-subspace lattice, abbreviated JSL. Every atomic Boolean subspace lattice, abbreviated ABSL, is a JSL. Any commutative JSL on Hilbert space, as well as any JSL on finite-dimensional space, is an ABSL. For any JSL L both Lat Alg L: and L-perpendicular to (on reflexive space) are JSL's. Those families of subspaces which arise as the set of atoms of some JSL on X are characterised in a way similar to that previously found for ABSL's. This leads to a definition of a subspace M-basis of X which extends that of a vector M-basis. New subspace M-bases arise from old ones in several ways. In particular, if {M gamma}(gamma is an element of Gamma) is a subspace M-basis of X, then (i) {(M-gamma')(perpendicular to)}(gamma is an element of Gamma) is a subspace M-basis of V-gamma is an element of Gamma(M-gamma')(perpendicular to), (ii) {K gamma}(gamma is an element of Gamma) is a subspace M-basis of V-gamma is an element of Gamma K-gamma for every family {K-gamma}(gamma is an element of Gamma) of subspaces satisfying (0) not equal K gamma subset of or equal to M gamma (gamma is an element of Gamma) and (iii) if X is reflexive, then {boolean AND(beta not equal gamma) M-beta'}(gamma)is an element of Gamma is a subspace M-basis of X. (Here M-gamma' is given by M-gamma' = V-beta not equal gamma M-beta).