论文标题
prime $ c^*$ - 代数的希尔伯特模块上的小学操作员
Elementary operators on Hilbert modules over prime $C^*$-algebras
论文作者
论文摘要
让$ x $成为$ c^*$ - 代数$ a $的右Hilbert模块，配备了规范操作员空间结构。我们将$ x $上的基本运算符定义为地图$ ϕ：x \ to x $，其中存在有限数量的元素$ u_i $中的$ c^*$ - algebra $ \ algebra $ \ mathbb {b}（x）$ auskoint-a相可靠的运算符$ x $ x $的$ x $ and $ v_i $ umelgbra $ ungebra $ ungebra $ sum umm a（a） u_i xv_i $ for $ x \ in x $。如果$ x = a $，此概念与$ a $的基本操作员的标准概念一致。在本文中，我们通过表明每个基本操作员的完全有限的标准在非零的hilbert $ a $ a $ a $ a $ a -module $ x $上与haagerup标准相符的haagerup规范，以$ \ mathbb {b} $ a $ a $ ifime a $ ifime a a $ ifime an $ c^*$ - 代数。
Let $X$ be a right Hilbert module over a $C^*$-algebra $A$ equipped with the canonical operator space structure. We define an elementary operator on $X$ as a map $ϕ: X \to X$ for which there exists a finite number of elements $u_i$ in the $C^*$-algebra $\mathbb{B}(X)$ of adjointable operators on $X$ and $v_i$ in the multiplier algebra $M(A)$ of $A$ such that $ϕ(x)=\sum_i u_i xv_i$ for $x \in X$. If $X=A$ this notion agrees with the standard notion of an elementary operator on $A$. In this paper we extend Mathieu's theorem for elementary operators on prime $C^*$-algebras by showing that the completely bounded norm of each elementary operator on a non-zero Hilbert $A$-module $X$ agrees with the Haagerup norm of its corresponding tensor in $\mathbb{B}(X)\otimes M(A)$ if and only if $A$ is a prime $C^*$-algebra.
