In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded continuous functions on X, whose real parts are dense in the algebra of bounded continuous real functions on X. The concept was introduced by Template:Harvs.
Let be the set of all rational functions that are continuous on ; in other words functions that have no poles in . Then
is a *-subalgebra of , and of . If is dense in , we say is a Dirichlet algebra.
It can be shown that if an operator has as a spectral set, and is a Dirichlet algebra, then has a normal boundary dilation. This generalises Sz.-Nagy's dilation theorem, which can be seen as a consequence of this by letting
- Completely Bounded Maps and Operator Algebras Vern Paulsen, 2002 ISBN 0-521-81669-6