Topological divisors of zero and Shilov boundary
Let \(L\) be a non-archimedean and non-trivially complete valued field, let \((A, \|. \| )\) be a commutative normed \(L\)-algebra with unity, and let \(\|.\| _{si}\) be the spectral seminorm of \(A\) defined by \(\| x \| _{si}:= \lim_{n} \| x^n \| ^{1/n}\). The Shilov boundary of \(A\) is the smallest among all the subsets \(F\) of Mult\((A,\|. \| _{si})\) (the set of all non-zero multiplicative \(\| . \| _{si}\)-continuous seminorms on \(A\)) that satisfy the following properties: (a) \(F\) is closed (with respect to the topology of simple convergence on \(\text{Mult}(A,\|.\| _{si})\)); (b) for each \(x \in A\) there exists a \(\psi \in F\) such that \(\psi(x) = \| x \| _{si}\). Also, an \(f \in A\) is said to be a topological divisor of zero if there exists a sequence \((x_n)_n\) in \(A\) such that \(\inf_n \| x_n \| >0\) and \(\lim_{n} \| f x_n \| =0\)). The author gives the following description for the topological divisors of zero by means of the Shilov boundary: If \(\|.\| = \|.\| _{si}\), then an \(f \in A\) is a topological divisor of zero if and only if there exists a \(\psi\) in the Shilov boundary of \(A\) such that \(\psi(f) =0\). He also proves that if \(A\) is a Banach \(L\)-algebra and if \(f \in A\) is not a divisor of zero, then \(f\) is a topological divisor of zero if and only if the ideal \(f A\) is not closed. By using these results, together with other ones obtained by the author in some of his previous papers, he derives interesting properties for the topological divisors of zero of certain types of algebras. Among them, we point out the following : (1) Let \(A\) be a Noetherian Banach \(L\)-algebra (e.g., \(A\) is an \(L\)-affinoid algebra\()\). Then any topological divisor of zero is a divisor of zero. (2) Let \(K\) be a complete algebraically closed field extension of \(L\), let \(D\) be a closed bounded infraconnected subset of \(K\), let \(H(D)\) be the completion of the \(K\)-algebra of rational functions without poles in \(D\), equipped with the norm of uniform convergence on \(D\). Then, for an \(f \in H(D)\), \(f \neq 0\), the following are equivalent: \((\alpha)\) \(f\) is a topological divisor of zero, \((\beta)\) the ideal \(f H(D)\) is not closed, \((\gamma)\) \(f\) is not quasi-invertible (i.e., it does not admit a factorization in the form \(Pg\) where \(g\) is an invertible element of \(H(D)\) and \(P\) is a polynomial whose zeros lie in the interior of \(D\)).
scientific article; zbMATH DE number 5013531