# Dirichlet algebra

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.

## Example

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

## References

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*Completely Bounded Maps and Operator Algebras*Vern Paulsen, 2002 ISBN 0-521-81669-6- {{#invoke:citation/CS1|citation

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