Diskalgebra

The disc algebra (sometimes called disc algebra) is considered in the mathematical sciences functional analysis and function theory algebra. Many functional analytic properties of the disc algebra are direct consequences functional theoretical propositions.

Definition

Identifies the circular disc, so is the set of all continuous functions which are holomorphic in the interior.

The definitions

,

Which make a complex algebra with involution, the so-called disc algebra.

Apparently a Unteralgbra the function algebra of continuous functions. is with respect to the supremum norm, which makes it a Banach algebra, completed, for uniform limits of holomorphic functions are holomorphic again. is therefore itself a Banach algebra, even with isometric involution, that is, it applies to all. The disc algebra is also Unterbanachalgebra by, the Banach algebra of all. Holomorphic and bounded functions on the supremum

Means limited to the edge of one obtains a figure. This illustration is an isometric homomorphism according to the maximum principle for holomorphic functions. In this sense, one can also be regarded as Unterbanachalgebra of. is the set of all continuous functions holomorphic can continue after. This would be an alternative definition of the disc algebra.

The disc algebra of generated, that is, the smallest Unterbanachalgebra that contains this feature, the disc itself is algebra

The Gelfandraum

For each of the point evaluation is a homomorphism, and thus an element of the Gelfand space of the disc algebra. One can show that with all the disc algebra homomorphisms are with values ​​in the complex numbers already found, and that the mapping is a homeomorphism, where the so-called Gelfandtopologie is given by the relative weak -* topology. The Gelfandraum the disc algebra can therefore be identified with the circular disk. With this identification, the Gelfand transform is the identity on the disc algebra.

The non- regularity of the disc algebra

On a commutative Banach algebra Gelfandraum one considers the so-called skin-core topology, by the conclusion of surgery

Is given. If this together with the Gelfandtopologie, it is called the Banach algebra regularly. The disc algebra is an example of a non-regular Banach. In fact, the quantity is finished in the Gelfandtopologie in the identification. Is now, it follows for all, and it follows from the identity theorem for holomorphic functions. And therefore it follows with respect to the sheath-core topology, the latter can not therefore coincide with the Gelfandtopologie.

The Schilowrand

If we identify with, so the topological boundary coincides with the Shilov boundary. Purpose is to show that every function of the Diskalgbera that yes coincides with its Gelfand transform because of the identification made ​​, their maximum amount takes on the boundary of the disk, but that is exactly the statement of the maximum principle for holomorphic functions.