J-subspace lattices and subspace M-bases

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 LatAlg ℒ and ${\displaystyle {ℒ}^{\perp }}$ (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 ${\displaystyle {\left\{M_{\gamma }\right\}}_{\gamma \in \Gamma }}$ is a subspace M-basis of X, then (i) ${\displaystyle {\left\{{(M_{\gamma }\prime )}^{\perp }\right\}}_{\gamma \in \Gamma }}$ is a subspace M-basis of ${\displaystyle V_{\gamma \in \Gamma }^{M_{\gamma }\prime }^\perp }$, (ii) ${\displaystyle {\left\{K_{\gamma }\right\}}_{\gamma \in \Gamma }}$ is a subspace M-basis of ${\displaystyle V_{\gamma \in \Gamma }^K\_\gamma }$ for every family {K_γ}_{γ∈Γ}${\displaystyle of\subset spacessatiy\in g}$(0)≠ K_γ⊆M_γ(γ ∈Γ)${\displaystyle \text{and}\left(iii\right)ifXisref\le \xi ve,then}${⋂_{β ≠ γ}^M_β'}_{γ∈Γ}${\displaystyle isa\subset spaceM-basisofX.(Here}$M_γ'${\displaystyle isgivenby}$M_γ' = V_{β ≠ γ}^M_β!$!.)