Shilov boundary explained

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

\DeltalA

be its structure space equipped with the relative weak*-topology of the dual

{lA}^*

. A closed (in this topology) subset

\Delta{lA}

is called a boundary of

{lA}

\max_ |f(x)|=\max_ |f(x)|

for all

x\inlA

.The set

S = \bigcap\

is called the Shilov boundary. It has been proved by Shilov^[1] that

is a boundary of

{lA}

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

S\subset\DeltalA

which satisfies

is a boundary of

, and

is a boundary of

, then

S\subsetF

Let

D=\{z\in\Complex:|z|<1\}

{lA}=H^{infty(D)\cap{lC}(\bar{D})}

be the disc algebra, i.e. the functions holomorphic in

and continuous in the closure of

with supremum norm and usual algebraic operations. Then

\Delta{lA}=\bar{D}

and

S=\{|z|=1\}