A Groupoid Approach to C*-Algebras
A Groupoid Approach to C*-Algebras
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DOI：
10.1007/bfb0091072
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发表时间：
1980-05
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影响因子：
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通讯作者：
J. Renault
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文献类型：
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作者：
J. Renault
遍历理论和冯诺依曼代数理论之间的相互作用可以追溯到Murray和冯诺依曼通过群测度构造获得的非I型因子的例子[54]。最近，P.哈恩（P. Hahn）揭示了一个自然的、可能是决定性的观点，它把这两种理论结合起来。它使用了G. Mackey“揭示并利用了群论和遍历理论之间某些显然意义深远的类比”（[53]，第187页）。特别地，群测度代数可以看作是某个主测度广群的正则表示的冯·诺依曼代数。此外，代数的大多数性质可以用广群来解释。J. Feldman和C.摩尔[31]，在遍历等价关系的框架。此外，他们还抽象地刻画了由构造von Neumann代数而产生的代数。很自然地，期望拓扑局部紧广群在C*-代数理论中扮演类似的角色。拓扑和李群胚的概念是由埃雷斯曼引入的，用于微分拓扑和几何。最近对拓扑群胚的兴趣来自叶理理论（[10]，p. 273）。这似乎是微分几何的观点，而不是麦基的虚拟群的观点，引起了J.韦斯特曼的兴趣，在群胚，并导致他的建设卷积代数群胚，首先在传递（和局部平凡）的情况下[75]，然后在非传递的主要情况[77]。然而，与诱导表征理论的相关性在[75]中也是显而易见的。变换群的卷积代数已经使用了一段时间[16，37]。关于变换群C*-代数的主要工作，
The interplay between ergodic theory and von Neumann algebra theory goes back to the examples of non-type I factors which Murray and von Neumann obtained by the group measure construction [54]. A natural and probably definitive point of view which joins both theories has recently been exposed by P. Hahn [45]. It uses the notion of measure groupoid, introduced by G. Mackey" to bring to light and exploit certain apparently far reaching analogies between group theory and ergodic theory"([53], p. 187). In particular, the group measure algebra may be regarded as the von Neumann algebra of the regular representation of some principal measure groupoid. Moreover, most of the properties of the algebra may be interpreted in terms of the groupoid. The same standpoint is adopted by J. Feldman and C. Moore [31], in the framework of ergodic equivalence relations. Besides, they characterize abstractly the von Neumann algebras arising from their construction. It is natural to expect that topological locally compact groupoids play a similar role in the theory of C*-algebras. The notions of topological and of Lie groupoid were introduced by Ehresmann for applications to differential topology and geometry. More recent interest in topological groupoids has come from the theory of foliations ([10], p. 273). It seems to be the differential geometry point of view, rather than Mackey's virtual group point of view which aroused J. Westman's interest in groupoids and led him to the construction of convolution algebras of groupoids, first in the transitive (and locally trivial) case [75] and then in the non-transitive principal case [77]. However the relevance to the theory of induced representations is also apparent in [75]. Convolution algebras of transformation groups had already been used for some time [16, 37]. The main works about transformation group C*-algebras,