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Decomposability of finite rank operators in certain subspaces and algebras

Published online by Cambridge University Press:  17 April 2009

Jiankui Li
Jiankui Li
Affiliation:
Department of Mathematics, Hunan Normal University, Changsha, 410081, China Department of Mathematics, University of New Hampshire, Durham, NH 03824, United States of America, e-mail: [email protected]
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Let  be either a reflexive subspace or a bimodule of a reflexive algebra in B (H), the set of bounded operators on a Hilbert space H. We find some conditions such that a finite rank T ∈  has a rank one summand in  and  has strong decomposability. Let (ℒ) be the set of all operators on H that annihilate all the operators of rank at most one in alg ℒ. We construct an atomic Boolean subspace lattice ℒ on H such that there is a finite rank operator T in (ℒ) such that T does not have a rank one summand in (ℒ). We obtain some lattice-theoretic conditions on a subspace lattice ℒ which imply alg ℒ is strongly decomposable.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 2 , October 2001 , pp. 307 - 314
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Argyros, S., Lambrou, M.S. and Longstaff, W.E., ‘Atomic Boolean subspace lattices and applications to the theory of bases’, Mem. Amer. Math. Soc. 445 (1991), 94.Google Scholar
[2]Erdos, J.A., ‘Reflexivity for subspace maps and linear spaces of operators’, Proc. London Math. Soc. 53 (1986), 582–600.CrossRefGoogle Scholar
[3]Erdos, J.A. and Power, S.C., ‘Weakly closed ideals of nest algebras’, J. Operator Theory 7 (1982), 219–235.Google Scholar
[4]Han, D., ‘On -submodules for reflexive operator algebras’, Proc. Amer. Math. Soc. 104 (1988), 1067–1070.CrossRefGoogle Scholar
[5]Harrison, K.J. and Mueller, U.A., ‘Decomposability of reflexive cycle algebras’, J. London Math. Soc. 51 (1995), 148–160.CrossRefGoogle Scholar
[6]Hopenwasser, A. and Moore, R., ‘Finite rank operators in reflexive operators algebras’, J. London Math. Soc. 27 (1983), 331–338.CrossRefGoogle Scholar
[7]Katavolos, A. and Katsoulis, E., ‘Semisimplicity in operator algebras and subspaces’, J. London Math. Soc. 42 (1990), 365–372.CrossRefGoogle Scholar
[8]Katavolos, A., Lambrou, M.S. and Longstaff, W.E., ‘The decomposability of operators relative to two subspaces’, Studia Math. 105 (1993), 25–36.CrossRefGoogle Scholar
[9]Kraus, J. and Larson, D., ‘Reflexivity and distance formulae’, Proc. London Math. Soc. 53 (1986), 340–356.CrossRefGoogle Scholar
[10]Li, J., ‘Decomposability of certain reflexive algebras’, Houston J. Math. 23 (1997), 121–126.Google Scholar
[11]Longstaff, W.E., ‘Strongly reflexive lattices’, J. London Math. Soc. 11 (1975), 491–498.CrossRefGoogle Scholar
[12]Longstaff, W.E., ‘Operators of rank one in reflexive algebras’, Canad. J. Math. 28 (1976), 19–23.CrossRefGoogle Scholar
[13]Longstaff, W.E., Nation, J. and Panaia, O., ‘Abstract reflexive subspaces and completely distributive collapsibility’, Bull. Austral. Math. soc. 58 (1998), 245–260.CrossRefGoogle Scholar
[14]Longtsaff, W.E. and Panaia, O., ‘-subspaces and subspace M-bases’, Studia Math 139 (2000), 197–212.Google Scholar
[15]Panaia, O., Quasi-spatiality of isomorphisms for certain classes of operator algebras, Ph.D. Dissertation (University of Western Australia, 1995).Google Scholar
[16]Ringrose, J.R., ‘On some algebras of operators’, Proc. London Math. Soc. 15 (1965), 61–83.CrossRefGoogle Scholar