J-subspace lattices and subspace M-bases

Longstaff, W.; Panaia, Oreste

Studia Mathematica, Tome 141 (2000), p. 197-212 / Harvested from The Polish Digital Mathematics Library

Résumé

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 ${M_{γ}}_{γ \in Γ}$ is a subspace M-basis of X, then (i) ${(M_{γ}^{'})}^{⊥}_{γ \in Γ}$ is a subspace M-basis of $V_{γ \in Γ}^{(} M_{γ}^{'})^{⊥}$ , (ii) ${K_{γ}}_{γ \in Γ}$ is a subspace M-basis of $V_{γ \in Γ}^{K} γ$ 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 $.)$

Publié le : 2000-01-01

Zbl 0974.46016

EUDML-ID : urn:eudml:doc:216719

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     title = {J-subspace lattices and subspace M-bases},
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     year = {2000},
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Longstaff, W.; Panaia, Oreste. J-subspace lattices and subspace M-bases. Studia Mathematica, Tome 141 (2000) pp. 197-212. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv139i3p197bwm/

Bibliographie

[00000] [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). | Zbl 0738.47047

[00001] [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. | Zbl 0948.46007

[00002] [3] J. A. Erdos, Basis theory and operator algebras, in: Operator Algebras and Applications (Samos, 1996), A. Katavolos (ed.), Kluwer, 1997, 209-223. | Zbl 0919.47036

[00003] [4] P. A. Fillmore and J. P. Williams, On operator ranges, Adv. Math. 7 (1971), 254-281.

[00004] [5] C. Foiaş, Invariant para-closed subspaces, Indiana Univ. Math. J. 21 (1972), 887-906. | Zbl 0259.47005

[00005] [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. | Zbl 0796.47033

[00006] [7] A. Katavolos, M. S. Lambrou and W. E. Longstaff, Pentagon subspace lattices on Banach spaces, J. Operator Theory, to appear.

[00007] [8] M. S. Lambrou, Approximants, commutants and double commutants in normed algebras, J. London Math. Soc. (2) 25 (1982), 499-512. | Zbl 0457.47009

[00008] [9] M. S. Lambrou and W. E. Longstaff, Some counterexamples concerning strong M-bases of Banach spaces, J. Approx. Theory 79 (1994), 243-259. | Zbl 0820.46002

[00009] [10] M. S. Lambrou and W. E. Longstaff, Non-reflexive pentagon subspace lattices, Studia Math. 125 (1997), 187-199. | Zbl 0887.47006

[00010] [11] W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (11) 2 (1975), 491-498. | Zbl 0313.47002

[00011] [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.

[00012] [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.

[00013] [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. | Zbl 0920.47005

[00014] [15] W. E. Longstaff and P. Rosenthal, On two questions of Halmos concerning subspace lattices, Proc. Amer. Math. Soc. 75 (1979), 85-86. | Zbl 0404.47004

[00015] [16] O. Panaia, Quasi-spatiality of isomorphisms for certain classes of operator algebras, Ph. D. dissertation, University of Western Australia, 1995.

[00016] [17] G. Szasz, Introduction to Lattice Theory, 3rd ed., Academic Press, New York, 1963. | Zbl 0126.03703

[00017] [18] P. Terenzi, Block sequences of strong M-bases in Banach spaces, Collect. Math. 35 (1984), 93-114. | Zbl 0583.46013

[00018] [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. | Zbl 0805.46018