Idempotents in an ultrametric Banach algebra
Let IK be a complete ultrametric field and let \(A\) be a unital commutative ultrametric Banach IK-algebra. Suppose that the multiplicative spectrum admits a partition in two open closed subsets. Then there exist unique idempotents \(u,\ v\in A\) such that \(\phi(u)=1, \ \phi(v)=0 \ \forall \phi \in U, \ \phi(u)=0 \ \phi(v)=1 \ \forall \phi \in V\). Suppose that IK is algebraically closed. If an element \(x\in A\) has an empty annulus \(r<|\xi-a|<s\) in its spectrum \(sp(x)\), then there exist unique idempotents \(u,\ v\) such that \(\phi(u)=1, \ \phi(v)=0\) whenever \( \phi(x-a)\leq r\) and \(\phi(u)=0, \ \phi(v)=1\) whenever \(\phi(x-a)\geq s\).