Bicommutants and ranges of derivations

Abstract

Let $V$ be a vector space over a field $F$, $V^*$ its dual space and $L(V)$ the algebra of all linear operators on $V$. For an operator $a\in L(V)$ let $a*$ be its adjoint acting on $V*$, and for a subset $R$ of $L(V)$ let $R"$ be its bicommutant. If $R$ is the subalgebra of $L(V)$ generated by an operator $a$, we prove that the set $Z:={b*: b\in R}"$ is contained in ${b*: b\in R"}$; moreover $Z$ is described. This inclusion is equality if $V$ as a module over the polynomial algebra $R=F[t]$ via $t\mapsto a$ is nice enough (say torsion, or injective, or if it contains a copy of $R$ as a direct summand). Further, under the same assumption about $V$ for any $b\in L(V)$, $b\in(a)"$ if and only if the derivations $d_a$ and $d_b$ satisfy $d_b(F(V))\subseteq d_a(F(V))$, where $F(V)$ is the set of all finite rank operators on $V$. The inclusion $d_b(L(V))\subseteq d_a(L(V))$ also holds under these conditions.