P-Adic Spaces with Strict Topologies as Topological Algebras

Date:

C. G. Petalas and A. K. Katsaras

Department of Mathematics, University of Ioannina, Greece

Abstract:

Let C_b(X) be the space of all bounded continuous functions from a Hausdorff zero-dimentional space X to a complete non-Archimedean valued field ${\Bbb K}$. We study C_b(X) as a topological algebra under the strict topologies $\beta_{0}, \beta , \beta_{1}, \beta_{u}$ and $\beta_{e}$. It is shown that each of the topologies $\beta_{0}, \beta , \beta_{1}$ and $\beta_{u}$ is locally solid and that the multiplication on C_b(X) is continuous for each of the topologies $\beta_{0}, \beta , \beta_{1}$ and $\beta_{u}$. We also give necessary and sufficient conditions for the algebra C_b(X) to be locally m-convex under the above topologies.