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2000 | 139 | 3 | 197-212
Tytuł artykułu

J-subspace lattices and subspace M-bases

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Abstrakty
EN
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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Twórcy
  • Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia, oreste@maths.uwa.edu.au
Bibliografia
  • [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).
  • [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.
  • [3] J. A. Erdos, Basis theory and operator algebras, in: Operator Algebras and Applications (Samos, 1996), A. Katavolos (ed.), Kluwer, 1997, 209-223.
  • [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.
  • [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.
  • [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.
  • [9] M. S. Lambrou and W. E. Longstaff, Some counterexamples concerning strong M-bases of Banach spaces, J. Approx. Theory 79 (1994), 243-259.
  • [10] M. S. Lambrou and W. E. Longstaff, Non-reflexive pentagon subspace lattices, Studia Math. 125 (1997), 187-199.
  • [11] W. E. Longstaff, Strongly reflexive lattices, J. London Math. Soc. (11) 2 (1975), 491-498.
  • [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.
  • [15] W. E. Longstaff and P. Rosenthal, On two questions of Halmos concerning subspace lattices, Proc. Amer. Math. Soc. 75 (1979), 85-86.
  • [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.
  • [18] P. Terenzi, Block sequences of strong M-bases in Banach spaces, Collect. Math. 35 (1984), 93-114.
  • [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.
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Bibliografia
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