论文标题

局部紧凑型组的痕迹

Traces on locally compact groups

论文作者

Forrest, Brian, Spronk, Nico, Wiersma, Matthew

论文摘要

我们对局部紧凑型组的痕迹进行系统研究,特别是在其通用和降低的C* - 代数上的痕迹。我们介绍痕量内核,并检查其与von Neumann内核和小型不变邻里(SIN)商的关系。在这样做的过程中,我们介绍了剩余的 - $ sin $组,其中包含$ sin $ and Maximimimimimimimimimimimimation Groups。我们详细检查了连接组的痕量内核。我们研究了还原的C* - 代数上的痕迹,为紧凑产生的组提供了一个简单的证据,这种痕迹的存在与拥有开放的正常amenable亚组相当,并且我们显示的非交流组承认独特的痕迹。 我们通过检查正式迹线和分解属性来结束。我们展示了与von Neumann内核一致的可及痕量核的财产(t)组。我们向完全断开的群体展示了可正常的痕量分离意味着分解属性。我们使用可及的痕迹来提供一个简单的证据,表明该组的合作性等同于同时核性,并具有其减少的C*-Algebra的痕迹。作为在本文中获得的结果的最终应用,我们将C* - 代数组嵌入到简单的AF代数中。结果,如果局部紧凑的基团是可正常的且奇特的分离的(痕量核很微不足道),则其还原的c*-Algebra为quasi-diagonal。

We conduct a systematic study of traces on locally compact groups, in particular traces on their universal and reduced C*-algebras. We introduce the trace kernel, and examine its relation to the von Neumann kernel and to small-invariant neighbourhood (SIN) quotients. In doing so, we introduce the class of residually-$SIN$ groups, which contains both $SIN$ and maximally almost periodic groups. We examine in detail the trace kernel for connected groups. We study traces on reduced C*-algebras, giving a simple proof for compactly generated groups that existence of such a trace is equivalent to having an open normal amenable subgroup, and we display non-discrete groups admitting unique trace. We finish by examining amenable traces and the factorization property. We show for property (T) groups that amenable trace kernels coincide with von Neumann kernels. We show for totally disconnected groups that amenable trace separation implies the factorization property. We use amenable traces to give a simple proof that amenability of the group is equivalent to simultaneous nuclearity and possessing a trace of its reduced C*-algebra. As a final application of the results obtained in the paper, we address the embeddability of group C*-algebras into simple AF algebras. As a consequence, if a locally compact group is amenable and tracially separated (trace kernel is trivial), then its reduced C*-algebra is quasi-diagonal.

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