Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-23T02:54:31.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Abstract reflexive sublattices and completely distributive collapsibility

Published online by Cambridge University Press:  17 April 2009

W. E. Longstaff ,
J. B. Nation  and
Oreste Panaia
W. E. Longstaff
Affiliation:
Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia e-mail: [email protected], [email protected]
J. B. Nation
Affiliation:
Department of Mathematics, The University of Western Australia, Nedlands, WA 6907, Australia e-mail: [email protected], [email protected]
Oreste Panaia
Affiliation:
Department of Mathematics, University of Hawaii, Honolulu, HI 96822-2273, United States of America e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

There is a natural Galois connection between subspace lattices and operator algebras on a Banach space which arises from the notion of invariance. If a subspace lattice ℒ is completely distributive, then ℒ is reflexive. In this paper we study the more general situation of complete lattices for which the least complete congruence δ on ℒ such that ℒ/δ is completely distributive is well-behaved. Our results are purely lattice theoretic, but the motivation comes from operator theory.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 58 , Issue 2 , October 1998 , pp. 245 - 260
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bandelt, H.-J., ‘On complete distributivity and maximal d-intervals in complete lattices’, in Colloq. Math. Soc. János Bolyai 14, Lattice Theory (North-Holland Publishing Co., Amsterdam, 1976), pp. 29–43.Google Scholar
[2]Bandelt, H.-J., ‘Tight residuated mappings and d-extensions’, in Colloq. Math. Soc. János Bolyai 29, Universal Algebra (North-Holland Publishing Co., Amsterdam, 1982), pp. 61–72.Google Scholar
[3]Freese, R. and Nation, J.B., ‘Covers in free lattices’, Trans. Amer. Math. Soc. 288 (1985), 1–42.Google Scholar
[4]Halmos, P.R., ‘Reflexive lattices of subspaces’, J. London Math. Soc. 4 (1971), 257–263.Google Scholar
[5]Hopenwasser, A., ‘Complete distributivity’, in Operator theory: Operator algebras and applications, Proc. Symp. Pure Math. 51 (Amer. Math. Soc, Providence, RI, 1990), pp. 285–305.Google Scholar
[6]Katavolos, A. and Katsoulis, E., ‘Semisimplicity in operator algebras and subspace lattices’, J. London Math. Soc. 42 (1990), 365–372.Google Scholar
[7]Longstaff, W.E., ‘Strongly reflexive lattices’, J. London Math. Soc. 1 (1975), 491–498.Google Scholar
[8]Longstaff, W.E., ‘Remarks on semisimple reflexive algebras’, in Proc. Conf. Automatic Continuity and Banach Algebras, (Loy, R.J., Editor), Centre for Math. Anal. 21 (A.N.U., Canberra, Australia, 1989), pp. 273–287.Google Scholar
[9]Longstaff, W.E., ‘A note on the semisimplicity of reflexive operator algebras’, in Proc. Int. Workshop on Analysis and its Applic., 4th Annual Meeting (Dubrovnik-Kupari, Yugoslavia, 1990), pp. 45–50.Google Scholar
[10]Longstaff, W.E. and Rosenthal, P., ‘On two questions of Halmos concerning subspace lattices’, Proc. Amer. Math. Soc. 75 (1979), 85–86.Google Scholar
[11]Panaia, O., Quasi-spatiality of isomorphisms for certain classes of operator algebras, (Ph.D. Thesis) (University of Western Australia, 1995).Google Scholar
[12]Raney, G.N., ‘Completely distributive complete lattices’, Proc. Amer. Math. Soc. 3 (1952), 677–680.Google Scholar
[13]Raney, G.N., ‘Tight Galois connections and complete distributivity’, Trans. Amer. Math. Soc. 97 (1960), 418–426.Google Scholar