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 be a commutative Banach algebra and let be its structure space equipped with the relative weak*-topology of the dual . A closed (in this topology) subset of is called a boundary of if for all . The set is called the Shilov boundary. It has been proved by Shilov[1] that is a boundary of .

Thus one may also say that Shilov boundary is the unique set which satisfies

  1. is a boundary of , and
  2. whenever is a boundary of , then .

Examples

be the disc algebra, i.e. the functions holomorphic in and continuous in the closure of with supremum norm and usual algebraic operations. Then and .

References

Notes

  1. Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: Functional analysis and semigroups. -- AMS, Providence 1957.

See also

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