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 Andrew Gleason (1957).

Example

Let ${\mathcal {R}}(X)$ be the set of all rational functions that are continuous on $X$ ; in other words functions that have no poles in $X$ . Then

{\mathcal {S}}={\mathcal {R}}(X)+{\overline {{\mathcal {R}}(X)}}

is a *-subalgebra of $C(X)$ , and of $C\left(\partial X\right)$ . If ${\mathcal {S}}$ is dense in $C\left(\partial X\right)$ , we say ${\mathcal {R}}(X)$ is a Dirichlet algebra.

It can be shown that if an operator $T$ has $X$ as a spectral set, and ${\mathcal {R}}(X)$ is a Dirichlet algebra, then $T$ has a normal boundary dilation. This generalises Sz.-Nagy's dilation theorem, which can be seen as a consequence of this by letting

X=\mathbb {D} .

References

Gleason, Andrew M. (1957), "Function algebras", in Morse, Marston; Beurling, Arne; Selberg, Atle (eds.), Seminars on analytic functions: seminar III : Riemann surfaces; seminar IV : theory of automorphic functions; seminar V : analytic functions as related to Banach algebras, vol. 2, Institute for Advanced Study, Princeton, pp. 213–226, Zbl 0095.10103
Nakazi, T. (2001) [1994], "Dirichlet algebra", Encyclopedia of Mathematics, EMS Press
Completely Bounded Maps and Operator Algebras Vern Paulsen, 2002 ISBN 0-521-81669-6
Wermer, John (November 2009), Bolker, Ethan D. (ed.), "Gleason's work on Banach algebras" (PDF), Andrew M. Gleason 1921–2008, Notices of the American Mathematical Society, 56 (10): 1248–1251.

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