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Commutants and double commutants of reflexive algebras. (English) Zbl 0904.47035
The author studies the commutant and the double commutant of the algebra $$\text{alg }{\mathcal L}$$ of all bounded operators on a Banach space $$X$$ leaving invariant each member of a lattice $${\mathcal L}$$ of subspaces of $$X$$. For example, he proves that when $${\mathcal L}$$ is the pentagon subspace lattice, then the only operators commuting with all the operators in $$\text{alg }{\mathcal L}$$ are multiples of the identity, hence $$(\text{alg }{\mathcal L})''= L(X)$$.
He also gives an example of an algebra $${\mathcal A}$$ of operators on a complex Hilbert space which is reflexive ($${\mathcal A}=\text{alg}(\text{lat }{\mathcal A}$$)), whose commutant and double commutant coincide, but are not reflexive.
##### MSC:
 47L30 Abstract operator algebras on Hilbert spaces
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