On isomorphisms and hyper-reflexivity of closed subspace lattices (Q1178779)

A subalgebra \({\mathcal A}\) of bounded linear operators on a complex Hilbert space \({\mathcal H}\) is said to be reflexive if \(\text{Alg Lat }{\mathcal A}={\mathcal A}\), where \(\text{Lat }{\mathcal A}\) denotes the lattice of closed subspaces of \({\mathcal H}\) invariant for every operator in \({\mathcal A}\), and \(\text{Alg }{\mathcal F}\) denotes the set of all operators in \({\mathcal H}\) leaving invariant the subspaces of a lattice \({\mathcal F}\). Dually a lattice of subspaces \({\mathcal F}\) is said to be reflexive if \(\text{Lat Alg }{\mathcal F}={\mathcal F}\). A reflexive algebra \({\mathcal A}\) is called hyper-reflexive if there exists a constant \(K\) such that for every bounded linear operator on \({\mathcal H}\), \(\sup\{\| P^ \perp TP\|\), \(P\in\text{Lat }{\mathcal A}\}\leq\text{dist}(T,{\mathcal A})\leq K\sup\{\| P^ \perp TP\|\), \(P\in\text{Lat }{\mathcal A}\}\) being \(P\) a projection, or its rank. Hyper-reflexive algebras were introduced by Arveson, generalizing the notion of nested algebras. For the latter the constant is 1. A lattice of subspaces \({\mathcal F}\) is called hyper-reflexive if \({\mathcal F}\) is reflexive and \(\text{Alg }{\mathcal F}\) is hyper-reflexive. In the present paper the author proves that lattice hyper-reflexivity is invariant through lattice isomorphisms, and also through lattice homomorphisms of a certain kind (namely those induced by an operator with minimum modulus greater than zero). There are several misprintings, which the reader can easily detect.