请输入您要查询的百科知识：

　

词条 Shilov boundary
释义

  1. Precise definition and existence

  2. Examples

  3. References

  4. Notes

  5. See also

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

  • Let be the open unit disc in the complex plane and let

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

  • {{Springer|id=B/b110310|title=Bergman-Shilov boundary}}

Notes

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

See also

  • James boundary
  • Furstenberg boundary

1 : Banach algebras
随便看

　

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

　

Copyright © 2023 OENC.NET All Rights Reserved
京ICP备2021023879号 更新时间：2024/9/21 0:44:26