SHILOV BOUNDARY FOR NORMED ALGEBRAS

摘要

Let E be a field provided with an absolute value for which it is complete and let A be a commutative S-algebra with unity provided with a semi-multiplicative (or power multiplicative) E-algebra semi-norm ||·||. Let Mult(A, ||·||) be the set of multiplicative -E-algebra semi-norms continuous with respect to ||·||. We show the existence of a Shilov boundary for ||·||, i.e. a closed subset F of Mult(A, ||·||), minimal for inclusion, such that for every x ∈ A, there exists Φ ∈ F such that Φ(x) = ||x||. In particular, if the field E is ultrametric, it applies to the spectral semi-norm of an ultrametric S-algebra.