Tree structure on the set of multiplicative semi-norms of Krasner algebras \(H(D)\) (Q5926244)

Let \(K\) be an algebraically closed complete field (complete with respect to an ultra metric absolute value) and \(D\) be an infinite subset of \(K\). Let \(H(D)\) denote the set of all analytic elements of \(D\), namely the completion of \(R(D)\), the \(K\)-subalgebra of \(K^D\) of rational functions with no poles in \(D\), with the topology \({\mathcal U}_D\) of uniform convergence on \(D\). Denote by \(S= \text{Mult}(H(D),{\mathcal U}_D)\) the set of seminorms \(\psi\) of the \(K\)-vector space \(H(D)\) which are continuous and which satisfy \(\psi(fg)= \psi(f)\psi(g)\) whenever \(f\), \(g\), \(fg\in H(D)\). Using a suitable topology of \(\tau\) for \(S\) it is established that the connectedness, the arc-connectedness of \(S\) are equivalent to the infra connectedness of \(D\). If \(D\subset K\) is closed and bounded and \(S\) is countable then the topology \(\tau\) for \(S\) is shown to be metrizable.