标题： 群的中心合并与RFD性质 显示英文标题

标题： Central amalgamation of groups and the RFD property

摘要： 研究哪些离散群G的全群C*-代数C*(G)是剩余有限维的（RFD）是一个古老而具有挑战性的问题。特别是，关于RFD性质在基本构造（如合并自由积和HNN扩张）下的行为了解不多。在[CS19]中证明了虚拟阿贝尔群的中心合并自由积是RFD的。在本文中，我们证明这一结论远超这种情况。我们的方法是通过展示从中心子群诱导的特征的某种逼近性质。特别是，这使我们能够证明在有限生成中心子群上合并的可解有限群的自由积是RFD的。另一方面，我们证明了RFD C*-代数（和群）的类并不在中心合并自由积下闭合。即我们给出了一个RFD群的例子（事实上是有限生成的可换RF群），其中心合并自由积不是RFD的，而且甚至不是最大几乎周期的。这回答了Khan和Morris [KM82]提出的问题。 显示英文摘要

摘要： It is an old and challenging topic to investigate for which discrete groups G the full group C*-algebra C*(G) is residually finite-dimensional (RFD). In particular not much is known about how the RFD property behaves under fundamental constructions, such as amalgamated free products and HNN-extensions. In [CS19] it was proved that central amalgamated free products of virtually abelian groups are RFD. In this paper we prove that this holds much beyond this case. Our method is based on showing a certain approximation property for characters induced from central subgroups. In particular it allows us to prove that free products of polycyclic-by-finite groups amalgamated over finitely generated central subgroups are RFD. On the other hand we prove that the class of RFD C*-algebras (and groups) is not closed under central amalgamated free products. Namely we give an example of RFD groups (in fact finitely generated amenable RF groups) whose central amalgamated free product is not RFD, moreover it is not even maximally almost periodic. This answers a question of Khan and Morris [KM82].