J-subspace lattices and subspace M-bases

W. Longstaff; Oreste Panaia

Studia Mathematica (2000)

  • Volume: 139, Issue: 3, page 197-212
  • ISSN: 0039-3223

Abstract

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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 Kγγ∈Γ o f s u b s p a c e s s a t i s f y i n g (0)≠ Kγ⊆Mγ(γ ∈Γ) a n d ( i i i ) i f X i s r e f l e x i v e , t h e n ⋂β ≠ γMβ’γ∈Γ i s a s u b s p a c e M - b a s i s o f X . ( H e r e Mγ’ i s g i v e n b y Mγ’ = Vβ ≠ γMβ . )

How to cite

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Longstaff, W., and Panaia, Oreste. "J-subspace lattices and subspace M-bases." Studia Mathematica 139.3 (2000): 197-212. <http://eudml.org/doc/216719>.

@article{Longstaff2000,
abstract = {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) $\{(M_γ^\{\prime \})^⊥\}_\{γ∈Γ\}$ is a subspace M-basis of $V_\{γ∈Γ\}^(M_γ^\{\prime \})^⊥$, (ii) $\{K_γ\}_\{γ∈Γ\}$ is a subspace M-basis of $V_\{γ ∈Γ\}^K_γ$ for every family Kγγ∈Γ$ of subspaces satisfying $(0)≠ Kγ⊆Mγ(γ ∈Γ)$ and (iii) if X is reflexive, then $⋂β ≠ γMβ’γ∈Γ$ is a subspace M-basis of X. (Here $Mγ’$ is given by $Mγ’ = Vβ ≠ γMβ$.)$},
author = {Longstaff, W., Panaia, Oreste},
journal = {Studia Mathematica},
keywords = {-lattices; -subspace lattice},
language = {eng},
number = {3},
pages = {197-212},
title = {J-subspace lattices and subspace M-bases},
url = {http://eudml.org/doc/216719},
volume = {139},
year = {2000},
}

TY - JOUR
AU - Longstaff, W.
AU - Panaia, Oreste
TI - J-subspace lattices and subspace M-bases
JO - Studia Mathematica
PY - 2000
VL - 139
IS - 3
SP - 197
EP - 212
AB - 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) ${(M_γ^{\prime })^⊥}_{γ∈Γ}$ is a subspace M-basis of $V_{γ∈Γ}^(M_γ^{\prime })^⊥$, (ii) ${K_γ}_{γ∈Γ}$ is a subspace M-basis of $V_{γ ∈Γ}^K_γ$ for every family Kγγ∈Γ$ of subspaces satisfying $(0)≠ Kγ⊆Mγ(γ ∈Γ)$ and (iii) if X is reflexive, then $⋂β ≠ γMβ’γ∈Γ$ is a subspace M-basis of X. (Here $Mγ’$ is given by $Mγ’ = Vβ ≠ γMβ$.)$
LA - eng
KW - -lattices; -subspace lattice
UR - http://eudml.org/doc/216719
ER -

References

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