---
tag: JHQNT58FEH
kind: concept
name: Shilov boundary
msc: [46J20, 46J10]
sources:
  - url: https://en.wikipedia.org/wiki/Shilov_boundary
same_as:
  wikidata: Q1632516
  wikipedia: Shilov boundary
---

## Summary

Let $A$ be a commutative unital Banach algebra.
Its Shilov boundary is the smallest closed subset of the [[character space|M7JVFJ3MMX]] on which every [[Gelfand transform|VWNSHKJT92]] attains its maximum modulus.
It is an abstract version of the maximum modulus principle of complex analysis.
For the disc algebra it is the unit circle, and for $C(X)$ it is all of $X$.

## Explanations

### L1 human

A holomorphic function on the closed unit disc takes its largest absolute value on the boundary circle.
So to know how large such a function can be, it suffices to look at the circle.
The Shilov boundary is the smallest closed set with this property, defined for any commutative Banach algebra with unit.
For the disc algebra the circle cannot be made smaller: for each point of the circle there is a function in the algebra that peaks exactly there.