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.
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
is called a boundary of
max
max
The set
is called the Shilov boundary.
It has been proved by Shilov[1] that
Thus one may also say that Shilov boundary is the unique set
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.