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J-subspace lattices and subspace M-bases

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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 $ℒ^⊥$ (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 ${M_γ}_{γ∈Γ}$ is a subspace M-basis of X, then (i) ${(M_γ')^⊥}_{γ∈Γ}$ is a subspace M-basis of $V_{γ∈Γ}^(M_γ')^⊥$, (ii) ${K_γ}_{γ∈Γ}$ is a subspace M-basis of $V_{γ ∈Γ}^K_γ$ for every family {K_γ}_{γ∈Γ}$ of subspaces satisfying $(0)≠ K_γ⊆M_γ(γ ∈Γ)$ and (iii) if X is reflexive, then ${⋂_{β ≠ γ}^M_β'}_{γ∈Γ}$ is a subspace M-basis of X. (Here $M_γ'$ is given by $M_γ' = V_{β ≠ γ}^M_β$.)
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  • Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia
  • Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia
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