The set is called the Shilov boundary.

When "I" is the set of all indices, "M" is a single point and the boundary component is the Shilov boundary.

Some 'boundary' theories such as the collaring of an open manifold, Martin boundary, Shilov boundary and Fürstenberg 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.