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 $^*$. A closed (in this topology) subset $F$ of $\Delta$ is called a boundary of $$ if $\max_ , f(x), =\max_ , f(x),$ for all $x \in \mathcal A$.
The set $S = \bigcap\$ is called the Shilov boundary. It has been proved by ShilovTheorem 4.15.4 in Einar Hille, Ralph S. Phillips
Functional analysis and semigroups
-- AMS, Providence 1957. that $S$ is a boundary of $$.

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 $\mathbb D=\$ be the open unit disc in the complex plane and let $= H^\infty(\mathbb D)\cap (\bar)$ 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 $\Delta = \bar$ and $S=\$.

References

*

Notes

See also

* James boundary
* Furstenberg boundary

{{SpectralTheory

Banach algebras