# J-subspace lattices and subspace M-bases

Studia Mathematica (2000)

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

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topLongstaff, 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

top- [1] S. Argyros, M. S. Lambrou and W. E. Longstaff, Atomic Boolean subspace lattices and applications to the theory of bases, Mem. Amer. Math. Soc. 445 (1991). Zbl0738.47047
- [2] J. A. Erdos, M. S. Lambrou, and N. K. Spanoudakis, Block strong M-bases and spectral synthesis, J. London Math. Soc. 57 (1998), 183-195. Zbl0948.46007
- [3] J. A. Erdos, Basis theory and operator algebras, in: Operator Algebras and Applications (Samos, 1996), A. Katavolos (ed.), Kluwer, 1997, 209-223. Zbl0919.47036
- [4] P. A. Fillmore and J. P. Williams, On operator ranges, Adv. Math. 7 (1971), 254-281.
- [5] C. Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 21 (1972), 887-906. Zbl0259.47005
- [6] A. Katavolos, M. S. Lambrou and M. Papadakis, On some algebras diagonalized by M-bases of ${l}^{2}$, Integral Equations Oper. Theory 17 (1993), 68-94. Zbl0796.47033
- [7] A. Katavolos, M. S. Lambrou and W. E. Longstaff, Pentagon subspace lattices on Banach spaces, J. Operator Theory, to appear.
- [8] M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, J. London Math. Soc. (2) 25 (1982), 499-512. Zbl0457.47009
- [9] M. S. Lambrou and W. E. Longstaff, Some counterexamples concerning strong M-bases of Banach spaces, J. Approx. Theory 79 (1994), 243-259. Zbl0820.46002
- [10] M. S. Lambrou and W. E. Longstaff, Non-reflexive pentagon subspace lattices, Studia Math. 125 (1997), 187-199. Zbl0887.47006
- [11] W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (11) 2 (1975), 491-498. Zbl0313.47002
- [12] W. E. Longstaff, Remarks on semi-simple reflexive algebras, in: Proc. Conf. Automatic Continuity and Banach Algebras, R. J. Loy (ed.), Centre Math. Anal. 21, Austral. Nat. Univ., Canberra, 1989, 273-287.
- [13] W. E. Longstaff, A note on the semi-simplicity of reflexive operator algebras, Proc. Internat. Workshop Anal. Applic., 4th Annual Meeting (Dubrovnik-Kupari, 1990), 1991, 45-50.
- [14] W. E. Longstaff, J. B. Nation and O. Panaia, Abstract reflexive sublattices and completely distributive collapsibility, Bull. Austral. Math. Soc. 58 (1998), 245-260. Zbl0920.47005
- [15] W. E. Longstaff and P. Rosenthal, On two questions of Halmos concerning subspace lattices, Proc. Amer. Math. Soc. 75 (1979), 85-86. Zbl0404.47004
- [16] O. Panaia, Quasi-spatiality of isomorphisms for certain classes of operator algebras, Ph. D. dissertation, University of Western Australia, 1995.
- [17] G. Szasz, Introduction to Lattice Theory, 3rd ed., Academic Press, New York, 1963. Zbl0126.03703
- [18] P. Terenzi, Block sequences of strong M-bases in Banach spaces, Collect. Math. 35 (1984), 93-114. Zbl0583.46013
- [19] P. Terenzi, Every separable Banach space has a bounded strong norming biorthogonal sequence which is also a Steinitz basis, Studia Math. 111 (1994), 207-222. Zbl0805.46018

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