双同形物和衍生物的范围
Bicommutants and ranges of derivations
摘要：
设$V$是域$F$上的向量空间，$V^*$是它的对偶空间，$L(V)$是$V$上所有线性算子的代数.对于算子$a\in L(V)$，令$a*$为其作用于$V*$的伴随算子，对于$L(V)$的子集$R$，令$R"$为其双同变算子。若$R$是由算子$a$生成的$L(V)$的子代数，则证明了集合$Z:={b*: b\in R}"$包含在${b*: b\in R"}$中，并刻画了$Z$.如果$V$作为多项式代数$R=F[t]$上的模通过$t\mapsto a$足够好（比如扭转或内射，或者如果它包含$R$的副本作为直和项），则这种包含是相等的。进一步，在关于$V$的相同假设下，对于任何$b\in L(V)$，$b\in(a)"$当且仅当导子$d_a$和$d_b$满足$d_b(F(V))\subseteq d_a(F(V))$，其中$F(V)$是$V$上所有有限秩算子的集合。在这些条件下，包含$d_b(L(V))\subseteq d_a(L(V))$也成立。
英文摘要：
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.
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