On the bamboo-shoot topology of certain inductive limits of topological groups
Let \(\{(G_n,\tau_n),\phi_{n+1\: n}\}_{n\in \mathbb{N}}\) be an inductive system of topological groups, where \(\phi_{n+1\: n}:(G_n,\tau_n)\to(G_{n+1},\tau_{n+1})\) is a continuous homomorphism. It was shown by \textit{N. Tatsuuma, H. Shimomura} and \textit{T. Hirai} [J. Math. Kyoto Univ. 38, No. 3, 551-578 (1998; Zbl 0930.22002)] that the inductive limit topology \(\tau_{\text{ind}}\) on \(G =\varinjlim G_{n}\) is not necessarily a group topology but that under some mild conditions (the ``PTA condition'', in their terminology) a group topology can be defined on \(G\) so that every canonical map \(\phi_n :G_n \rightarrow G\) is continuous. They called the finest among such topologies the bamboo-shoot topology, \(\tau_{X_{\text{OPEN}}}\) for short. The present paper studies the topology \(\tau_{X_{\text{OPEN}}}\) for inductive systems of groups obtained from inductive limits of Banach algebras. These groups can be described in a concrete fashion as follows. Let \(\{A_ n, \psi_{n+1: n}\}_{n\in \mathbb{N}}\) be a strict inductive limit of Banach algebras with identity and assume further that its inductive limit \(A\) admits an identity. Next, let \(G(A_n)\) denote the group of invertible elements of \(A_n\) equipped with the norm topology of \(A_n\). Then \(\{G(A_n),\psi_{n+1 : n}\}_{n \in \mathbb{N}}\) is an inductive system of topological groups and its direct limit coincides, as a set, with the group of invertible elements of \(A\), \(G(A)\). On this set \(G(A)\) three topologies can be naturally brought into consideration: \(\tau_{X_{\text{OPEN}}}\) and \(\tau_{\text{ind}}\) arise when \(G(A)\) is regarded as an inductive limit of topological groups and \(\tau_{\text{lct}}\) arises from regarding \(G(A)\) as a subset of the locally convex algebra \(A\) (recall that \(A=\varinjlim A_{n}\)). The main results of this paper prove that \(\tau_{X_{\text{OPEN}}}\) coincides with \(\tau_{\text{lct}}\) on \(G(A)\) and that \(\tau_{\text{ind}}\) is a group topology if and only if all \(A_n\) are finite-dimensional. The last section deals with the case when \(A\) has no identity. A related inductive system is then defined so that its limit has a unit and the preceding results are then applied to this system.
scientific article; zbMATH DE number 1471429