On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms
The present paper is organized in two parts. Part I consists of Sections 1-5 and its purpose is to discuss various group topologies on inductive limits of topological groups, and unitary representations of inductive limit groups in a certain case. In Section 1 the authors give a counterexample and discuss some general properties of group topologies, especially for inductive limit groups. In Section 2, the authors introduce the notion of PTA-group, and construct in the category of PTA-groups, a group topology \(\tau_{BS}\), called Bamboo-Shoot (=BS) topology, on the limit group of a countable inductive system, and show that \(\tau_{BS}\) is the strongest among the group topologies weaker than \(\tau_{ind}\). In Section 3, the properties of BS-topology and in Section 4 a generalization of BS-topology to uncountable inductive systems are discussed. In Section 5, positive definite functions and unitary representations of the limit groups of countable LCG inductive systems are treated. In Part II, which consists of Sections 6-9, the authors discuss the case of the group \(G=\text{Diff}_0(M)\) of diffeomorphisms with compact supports on a connected, non-compact, \(C^r\)-manifold \(M\), \(1\leq r\leq\infty\), and prove that the inductive limit topology \(\tau_{ind}\) never gives a group topology (Theorem 6.1).
scientific article; zbMATH DE number 1309661