The local structure of twisted covariance algebras
The local structure of twisted covariance algebras
复制标题
扭曲协方差代数的局部结构
DOI：
10.1007/bf02392308
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发表时间：
1978
期刊：
影响因子：
3.7
通讯作者：
Philip Green
中科院分区：
文献类型：
--
作者：
Philip Green
研究可分局部紧群 G 的酉表示论的基本问题是确定其不可约表示（的等价类）的空间 G ̂ 。已知当G不是I型时，具有Mackey Borel结构的G ̂ 不是标准的，甚至是可数分离的。这通常被解释为意味着这样一个群的不可约表示是不可分类的，因此问题就变成了寻找 G ̂ 的替代品，足够简单以提供它可以被完整描述的希望，但又足够复杂以反映 q 表示论的重要部分。已经提出了两个有希望的候选者，两者都使用群 C* 代数 C*(G) 定义（与 G 具有相同的表示理论）：C*(G) 的原始理想的空间 Prim G，由 Effros [19] 证明是由壳核拓扑生成的 Borel 结构中的标准 Borel 空间；以及 C*(G) 的正态表示（可追踪因子表示）的准等价类的空间 Gau r，由 Halpern [35] 证明是 Mackey Borel 结构中的标准。 （[19] 和 [35] 的结果实际上对任意可分离的 C* 代数有效，而不仅仅是那些由群产生的代数。）在 G 是类型 I 的情况下，这两个空间都可以自然地识别为 G ̂ 。证明它们是自然研究对象的引人注目的证据可以在 Pukanszky 的美丽结果 [49] 中找到，即对于连通 G，它们是“相同的”，因为与 Gno 的任何元素或其成员的内核相关联的映射是 Gnor 到 Prim C*(G) 上的双射。 （很容易证明这种双相实际上是 Borel 同构。）
The fundamental problem in investigating the unitary representation theory of a separable locally compact group G is to determine its space G ̂ of (equivalence classes of) irreducible representations. I t is known that when G is not type I, G ̂ , with the Mackey Borel structure, is not standard, or even countably separated. This is generally interpreted to mean that the irreducible representations of such a group are not classifiable, and so the problem becomes to find a substitute for G ̂ , simple enough to afford some hope that it can be described completely, yet complicated enough to reflect a significant part of the representation theory of q. Two promising candidates have been proposed, both defined using the group C*-algebra C*(G) (which has the same representation theory as G): the space Prim G of primitive ideals of C*(G), which was shown by Effros [19] to be a standard Borel space in the Borel structure generated by the hull-kernel topology; and the space Gao r of quasi-equivalence classes of normal representations (traceable factor representations) of C*(G), shown by Halpern [35] to be standard in the Mackey Borel structure. (The results of [19] and [35] are actually valid for arbitrary separable C*-algebras, not just those arising from groups.) In the case that G is type I, both of these spaces may be naturally identified with G ̂ . Striking evidence that they are natural objects of s tudy may be found in the beautiful result [49] of Pukanszky, that for connected G they are "the same" in the sense that the map which associates to any element of Gno r the kernel of its members is a bijection of Gnor onto Prim C*(G). (It is easily shown that this bijoction is in fact a Borel isomorphism.)