Shilov boundary

In functional analysis, 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 $𝒜$ be a commutative Banach algebra and let $\mathrm{\Delta }𝒜$ be its structure space equipped with the relative weak*-topology of the dual ${𝒜}^{*}$. A closed (in this topology) subset $F$ of $\mathrm{\Delta }𝒜$ is called a boundary of $𝒜$ if $\underset{f\in \mathrm{\Delta }𝒜}{\max }|f(x)|=\underset{f\in F}{\max }|f(x)|$ for all $x\in 𝒜$. The set $S=\bigcap \{F:F\text{ is a boundary of }𝒜\}$ is called the Shilov boundary. It has been proved by Shilov that $S$ is a boundary of $𝒜$.

Thus one may also say that Shilov boundary is the unique set $S\subset \mathrm{\Delta }𝒜$ which satisfies

1. $S$ is a boundary of $𝒜$, and
2. whenever $F$ is a boundary of $𝒜$, then $S\subset F$.

Examples

Let $𝔻=\{z\in ℂ:|z|<1\}$ be the open unit disc in the complex plane and let $𝒜=H^{\mathrm{\infty }}(𝔻)\cap 𝒞(\overline{𝔻})$ be the disc algebra, i.e. the functions holomorphic in $𝔻$ and continuous in the closure of $𝔻$ with supremum norm and usual algebraic operations. Then $\mathrm{\Delta }𝒜=\overline{𝔻}$ and $S=\{|z|=1\}$.

References

• Template:Springer

See also

• James boundary
• Furstenberg boundary