Property T and Approximation of Operators
Property T and Approximation of Operators
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DOI：
10.1112/blms/22.1.25
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发表时间：
1990
影响因子：
0.9
通讯作者：
D. Voiculescu
中科院分区：
文献类型：
--
作者：
D. Voiculescu
用不同的方法在[6]和[7]中证明了有限维C*-代数算子的非正则极限的块对角算子（即有限秩算子的直接和）的存在。然而，正如SJ Szarek所指出的那样，这留下了展示这种非近似块对角算子的具体例子的问题，这是本注释的结果之一。实际上，与[7]中的方法相关，我们给出了具有足够多有限维表示的有限生成C*-代数的具体例子，这些有限生成C*-代数不能嵌入到任何其他C*-代数中。证明了无限维可分Hilbert空间上块对角算子的酉轨道闭包集的不可分性。特别是，任意算子的酉轨道的闭包形成了一个不可分的度量空间，从而回答了DA Herrero给我们的一个问题。本文的两个结果都是用性质为T的离散群的有限维表示的直接和得到的，例如SL （3， Z）。考虑到一个标准的矩阵技巧，建立我们的断言对于操作符的元组而不是单个操作符就足够了。如果Jf是复希尔伯特空间&{&)，则Jf （Ji^）和U {^ e)分别表示3tf上的有界算子、紧算子和酉算子。如果T =(我,……， n) e (J？（3f）) n是一个n元组的运算符，那么我们将使用范数|| T||= max^^|| 2J|| w元组的和和直接和将是组件式的。一个“-元”Te (J5？（> f）) n为对角线，如果T= TX©T2©…其中xj （jeN）是作用于有限维空间的算子的^-元组。拥有以下特别定义将会很有用。
It has been shown by different methods in [6] and [7] that there exist blockdiagonal operators (that is, direct sums of finite-rank operators) which are not normlimits of operators with finite-dimensional C*-algebras. This, however, as pointed out by SJ Szarek, left open the question of exhibiting concrete examples of such nonapproximable block-diagonal operators, which is one of the results of this note. Actually, related to the approach in [7], we produce concrete examples of finitely generated C*-algebras with sufficiently many finite-dimensional representations, which cannot be nuclearly embedded into any other C*-algebras. The other fact we prove is the nonseparability of the set of closures of unitary orbits of block-diagonal operators on a separable infinite-dimensional Hilbert space. In particular, the closures of unitary orbits of arbitrary operators form a nonseparable metric space, thus answering a question communicated to us by DA Herrero.Both results in this note are obtained using direct sums of finite-dimensional representations of a discrete group with property T, for instance SL (3, Z). In view of a standard matrix trick, it will be sufficient to establish our assertions for «-tuples of operators instead of single operators. If Jf is a complex Hilbert space & {&), Jf (Ji^) and U {^ e) will denote respectively the bounded operators, the compact and the unitary operators on 3tf. If T=(7i,..., Tn) e (J?(3f)) n is an n-tuple of operators then we will use the norm|| T||= max^^|| 2J|| and sums and direct sums of w-tuples will be componentwise. An «-tuple Te (J5?(> f)) n is block-diagonal if T= TX© T2©... where xj (jeN) are^-tuples of operators acting on finite-dimensional spaces. It will be useful to have the following ad hoc definition.