A LARGE WEAK OPERATOR CLOSURE FOR THE ALGEBRA GENERATED BY TWO ISOMETRIES
A LARGE WEAK OPERATOR CLOSURE FOR THE ALGEBRA GENERATED BY TWO ISOMETRIES
复制标题
两个等距生成代数的大型弱算子闭包
DOI：
--
复制
发表时间：
2005
期刊：
影响因子：
--
通讯作者：
C. Read
中科院分区：
文献类型：
--
作者：
C. Read
给定希尔伯特空间H上的两个（或n个）等距，使得它们的值域相互正交，可以用它们生成一个C <$-代数。如果值域和为H，则这个C <$-代数，Cuntz C <$-代数On，在<$-同构下是唯一的。然而，如果我们忽略了在<$operations下的闭包，而仅仅考虑由这种等距生成的范数闭代数，那肯定不是唯一的;即使我们取了这样一个代数的弱算子拓扑（WOT）闭包，它仍然不是唯一的。在人们可能想象得到的可能的WOT闭包中，最大的是冯·诺依曼代数，如果碰巧生成的WOT闭代数实际上在埃尔米特共轭下是封闭的，那么将得到它。在本文中，我们表明，这种最大可能的情况下确实可以发生（一个问题，这是由戴维森），我们展示了对等距S 0，S1的不相交的范围，使WOT封闭代数生成的S 0和S1是整个B（H）。
Given two (or n) isometries on a Hilbert space H, such that their ranges are mutually orthogonal, one can use them to generate a C ⁄ -algebra. If the ranges sum to H, then this C ⁄ -algebra, the Cuntz C ⁄ -algebra On, is unique up to ⁄-isomorphism. However if we omit to close under the ⁄ operation and merely consider the norm closed algebra generated by such isometries, that is certainly not unique; even if we take the weak operator topology (WOT) closure of such an algebra, it is still not unique. Among the possible WOT closures that one might conceivably get, the largest is the von Neumann algebra that will be obtained if it should chance to happen that the WOT closed algebra generated is, in fact, closed under hermitian conjugation after all. In this paper we show that this largest possible case can indeed happen (a problem which was posed by Davidson); we exhibit pairs of isometries S0, S1 with disjoint ranges, such that the WOT closed algebra generated by S0 and S1 is the whole of B(H).