Shilov boundary

The Shilov boundary (after Georgi Shilov, according to the English transcription also Shilov boundary ) is a mathematical concept from the theory of commutative Banach algebras, . Thus, a version of the well known from the theory of functions maximum principle is applied to commutative Banach algebras.

Motivation

For simplicity we restrict ourselves to commutative algebras with unit element. Let a compact Hausdorff space and a subalgebra of the Banach algebra of continuous functions with the following properties:

• , That is the constant function includes 1
• , That is, separates the points of

We then say short, be a function algebra on.

A closed subset is called maximizing ( for ) if the following applies to all functions.

If, for example, the disc and the disc algebra, ie the algebra of all continuous functions which are holomorphic in the interior, it is because of the maximum principle of the theory of functions each closed subset that contains the edge, a maximizing quantity. In particular, the smallest maximizing quantity.

Shilov boundary for Funktionenalgebren

The example of the disc algebra generalizes to the following going back to Shilov set:

• Are a compact Hausdorff space and a function algebra on, so is the average of all -maximizing quantities for non-empty and maximizing again.

In particular, there exists a smallest maximizing quantity. This is called the Shilov boundary of the function algebra, common names are or.

Shilov boundary for commutative Banach algebras

Let be a commutative Banach algebra with identity -. The Gelfand - space is known to be a compact Hausdorff space and the Gelfand transform maps a function algebra on from. The Shilov boundary of the function algebra is called Shilov boundary of and also with or referred to.

Examples

• The Gelfand space of the disk algebra is the set of point evaluations and the mapping is a homeomorphism. One identified by this homeomorphism with so and it is.
• Be the bicylinder with radius. is generated by all polynomials in two variables sub - Banach algebra of. One can show that the Gelfand space of the set of point evaluations is for and that is a homeomorphism. So you can identify as above. Then one can show that. In this case, the Shilov edge is lower than the topological border of IN.
• Is a compact Hausdorff space and so is.

Comments

Is a commutative Banach algebra with unit - element, as for the Gelfand transform that. This follows directly from the definitions, since is a maximizing quantity of the function algebra. The Gelfand transform thus satisfy a maximum principle with respect to the Shilov - boundary.

As is well known applies to the spectrum of the formula. With respect to the edges of the spectra following formula applies.