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.
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Precise definition and existence
Let
Thus one may also say that Shilov boundary is the unique set
-
S is a boundary ofA , and - whenever
F is a boundary ofA , thenS ⊂ F .
Examples