Bicommutants and ranges of derivations

摘要

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.