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 $mathcal\; A$ be a commutative Banach algebra and let $Delta\; mathcal\; A$ be its structure space equipped with the relative weak*-topology of the dual $\{mathcal\; A\}^*$. A closed (in this topology) subset $F$ of $Delta\; \{mathcal\; A\}$ is called a boundary of $\{mathcal\; A\}$ if $max\_\{f\; in\; Delta\; \{mathcal\; A\; |x(f)|=max\_\{f\; in\; S\}\; |x(f)|$ for all $x\; in\; mathcal\; A$.The set is called the Shilov boundary. It has been proved by Shilov [Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: [http://www.ams.org/online_bks/coll31/coll31-chIV.pdf Functional analysis and semigroups] . -- AMS, Providence 1957.] that $S$ is a boundary of $\{mathcal\; A\}$.

Thus one may also say that Shilov boundary is the unique set $S\; subset\; Delta\; mathcal\; A$ which satisfies
#$S$ is a boundary of $mathcal\; A$, and
#whenever $F$ is a boundary of $mathcal\; A$, then $S\; subset\; F$.

Examples

*Let be the open unit disc in the complex plane and let be the disc algebra, i.e. the functions holomorphic in $mathbb\; D$ and continuous in the closure of $mathbb\; D$ with supremum norm and usual algebraic operations. Then and $S=\{|z|=1\}$.

References

See also

*James boundary