J-subspace lattices and subspace M-bases

• Autorzy: W. E. Longstaff , Oreste Panaia
• Języki publikacji: EN

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 $M_γ' = V_{β ≠ γ}^M_β$.)

Kategorie tematyczne

• 47C05: Operators in algebras
• 06C15: Complemented lattices, orthocomplemented lattices and posets
• 46B15: Summability and bases

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2000
• Tom: 139
• Numer: 3
• Strony: 197-212

Daty

wydano

2000

otrzymano

1998-02-11

Twórcy

autor

W. E. Longstaff

• Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia

autor

Oreste Panaia

• Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia

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.