论文标题
非交通多项式的换向因子和图像
Commutators and images of noncommutative polynomials
论文作者
论文摘要
令$ a $为代数，让$ f $为非稳定的非交易性多项式。在本文的第一部分中，我们考虑$ [a，a] $，$ a $中的换向器的线性跨度和跨度$ f（a）$之间的关系，$ f $ in $ a $的图像的线性跨度。特别是，我们表明$ [a，a] = a $表示span $ f（a）= a $。在第二部分中，我们为多项式图像建立了一些警告类型的结果。 For example, we show that if $C$ is a commutative unital algebra over a field $F$ of characteristic $0$, $A$ is the matrix algebra $M_n(C)$, and the polynomial $f$ is neither an identity nor a central polynomial of $M_n(F)$, then every commutator in $A$ can be written as a difference of two elements, each of which is a sum of $ f（a）$的$ 7788 $元素（如果$ c = f $是代数关闭的字段，则$ 4 $元素就足够了）。其他一些代数也获得了类似的结果，尤其是Hilbert Space $ h $上所有有界线性运算符的代数$ b（h）$。
Let $A$ be an algebra and let $f$ be a nonconstant noncommutative polynomial. In the first part of the paper, we consider the relationship between $[A,A]$, the linear span of commutators in $A$, and span$f(A)$, the linear span of the image of $f$ in $A$. In particular, we show that $[A,A]=A$ implies span$f(A)=A$. In the second part, we establish some Waring type results for images of polynomials. For example, we show that if $C$ is a commutative unital algebra over a field $F$ of characteristic $0$, $A$ is the matrix algebra $M_n(C)$, and the polynomial $f$ is neither an identity nor a central polynomial of $M_n(F)$, then every commutator in $A$ can be written as a difference of two elements, each of which is a sum of $7788$ elements from $f(A)$ (if $C=F$ is an algebraically closed field, then $4$ elements suffice). Similar results are obtained for some other algebras, in particular for the algebra $B(H)$ of all bounded linear operators on a Hilbert space $H$.
