Bicommutants and ranges of derivations

Abstract

Let V be a vector space, V* its dual space and L(V) the algebra of all linear operators on V. For an operator a ∈ L(V) let a* be its adjoint acting on V*, and for a subset R of L(V) let R″ be its bicommutant. If R is a left noetherian subalgebra of L(V), then {a*: a ∈ R}″ = {a*: a ∈ R″}. When R is singly generated R″ is described precisely. Further, for any two operators a, b ∈ L(V), b ∈ (a)″ if and only if the derivations d _a and d _b satisfy d _b (F(V)) ⊆ d _a (F(V)), where F(V) is the set of all finite rank operators on V. In this case the inclusion d _b (L(V)) ⊆ d _a (L(V)) also holds.

Keywords:

• bicommutant
• adjoint operator
• derivation
• torsion
• injective module

AMS Subject Classifications::

• 15A04
• 16S50
• 16W25
• 16U70
• 47A05
• 47B47

View correction statement:

Bicommutants and ranges of derivations

Acknowledgements

The author is grateful to Matej Brešar and to the anonymous referee for their corrections of the first draft of this article. He was supported in part by the Ministry of Science and Education of Slovenia.