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PENTAGON SUBSPACE LATTICES ON BANACH SPACES

A. KATAVOLOS, M.S. LAMBROU and W.E. LONGSTAFF
Journal of Operator Theory
Vol. 46, No. 2 (Fall 2001), pp. 355-380
Published by: Theta Foundation
Stable URL: http://www.jstor.org/stable/24715501
Page Count: 26
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PENTAGON SUBSPACE LATTICES ON BANACH SPACES
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Abstract

If K, L and M are (closed) subspaces of a Banach space X satisfying K ∩ M = (0), K ∨ L = X and L βŠ‚ M, then π“Ÿ = {(0), K, L, M, X} is a pentagon subspace lattice on X. If π“Ÿ1 and π“Ÿ2 are pentagons, every (algebraic) isomorphism Ο† : Alg π“Ÿ1 β†’ Alg π“Ÿ2 is quasi-spatial. The SOT-closure of the fin- ite rank subalgebra of Alg π“Ÿ is {T ∈ Alg π“Ÿ : T(M) βŠ† L}. On separable Hilbert space H every positive, injective, non-invertible operator A and every non-zero subspace M satisfying M ∩ Ran(A) = (0) give rise to a pentagon π“Ÿ(A; M). Alg π“Ÿ(A; M) and Alg π“Ÿ(B; N) are spatially isomorphic if and only if T Ran(A) = Ran(B) and T(M) = N for an invertible operator T ∈ B(H). If π’œ(A) is the set of operators leaving Ran(A) invariant, every isomorphism Ο† : π’œ(A) β†’ π’œ(B) is implemented by an invertible operator T satisfying T Ran(A) = Ran(B).

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