Multiplicative spectrum of ultrametric Banach algebras of continuous functions (Q712203)

Let \(K\) be an ultrametric complete field and \(E\) be an ultrametric space. Let \(A\) be the Banach \(K\)-algebra of bounded continuous functions from \(E\) to \(K\) and let \(B\) be the Banach \(K\)-algebra of bounded uniformly continuous functions from \(E\) to \(K\). This paper studies maximal ideals and continuous multiplicative seminorms on \(A\) and \(B\) using relations of stickiness and contiguousness on ultrafilters that are equivalence relations. Thus the maximal spectrum of \(A\) (respectively, of \(B\)) is in one-to-one correspondence with the set of equivalence classes with respect to stickiness (respectively, contiguousness). Among many other results, it is shown that the Shilov boundary of \(A\) (respectively, of \(B\)) is equal to the whole set of continuous multiplicative seminorms of \(A\) (respectively, \(B\)).