Non Integrally Closed Algebras H(D)

Let ( K , | . | ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq901.tif"/> be an algebraically closed complete ultrametric field. Let D be a closed bounded infraconnected set in K, and let H(D) be the Banach algebra of the analytic elements in D. Let f be a power series whose radius of convergence is r. If for some s ∈ ℕ^*, f^s has a radius of convergence strictly greater than r, and if f^s has less than s zeros in C(0, r) then f does not belong to H(d(0,r ^-)).

Let a be an integer, prime to the residue characteristic p of K if p ≠ 0. The function 1 + x q https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072614/3ce7aa8c-759e-436c-a799-e22f41c5a790/content/eq902.tif"/> does not belong to H(d(0,1^-)).

Assume D to have a T-filter F with a non empty beach or a T-filter with center and with diameter in |K|. Then H(D) is not integrally closed.

The article [E₆ ] by A. Escassut (to appear in Communications in Algebra) concerns the case of the integrally closed algebras H(D).