Bicommutants and ranges of derivations

TL;DR

This paper investigates the structure of bicommutants and derivations in the algebra of linear operators, providing conditions under which certain inclusions become equalities and characterizing derivations via finite rank operators.

Contribution

It establishes new relationships between bicommutants, adjoint operators, and derivations in the algebra of linear operators, with specific conditions for equality and characterization.

Findings

The set Z is contained in the double commutant of R.

Equality holds when V is a torsion or injective module over F[t].

Derivations are characterized by their action on finite rank operators.

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.