Shilov boundary
In functional analysis, a branch of mathematics, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds. It is named after its discoverer, Georgii Evgen'evich Shilov. Precise definition and existence. Let formula_1 be a commutative Banach algebra and let formula_2 be its structure space equipped with the relative weak*-topology of the dual formula_3. A closed (in this topology) subset formula_4 of formula_5 is called a boundary of formula_6 if formula_7 for all formula_8. The set formula_9 is called the Shilov boundary. It  has been proved by Shilov that formula_10 is a boundary of formula_6. Thus one may also say that Shilov boundary is the unique set formula_12 which satisfies formula_19 be the disc algebra, i.e. the functions holomorphic in formula_20 and continuous in the closure of formula_20 with supremum norm and usual algebraic operations. Then formula_22 and formula_23.