View source for Dirichlet algebra

{{Short description|Algebraic structure}}
In [[mathematics]], a '''Dirichlet algebra''' is a particular type of [[C*-algebra|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 {{harvs|first=Andrew|last=Gleason|authorlink=Andrew Gleason|year=1957|txt}}.

==Example==

Let <math>\mathcal{R}(X)</math> be the set of all [[rational function]]s that are continuous on <math>X</math>; in other words functions that have no [[Pole (complex analysis)|poles]] in <math>X</math>. Then

:<math>\mathcal{S} = \mathcal{R}(X) + \overline{\mathcal{R}(X)}</math>

is a *-subalgebra of <math>C(X)</math>, and of <math>C\left(\partial X\right)</math>. If <math>\mathcal{S}</math> is [[Dense set|dense]] in <math>C\left(\partial X\right)</math>, we say <math>\mathcal{R}(X)</math> is a '''Dirichlet algebra'''.

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

:<math>X=\mathbb{D}.</math>

==References==

*{{Citation | last1=Gleason | first1=Andrew M. | authorlink = Andrew Gleason | editor1-last=Morse | editor1-first=Marston | editor2-last=Beurling | editor2-first=Arne | editor3-last=Selberg | editor3-first=Atle | title=Seminars on analytic functions: seminar III : Riemann surfaces; seminar IV : theory of automorphic functions; seminar V : analytic functions as related to Banach algebras | publisher=Institute for Advanced Study, Princeton | zbl=0095.10103 | year=1957 | volume=2 | chapter=Function algebras | pages=213–226}}
*{{eom|id=d/d120180|title=Dirichlet algebra|first=T. |last=Nakazi}}
*''Completely Bounded Maps and Operator Algebras'' Vern Paulsen, 2002 {{ISBN|0-521-81669-6}}
*{{citation|last=Wermer|first=John|title=Gleason's work on Banach algebras|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1248–1251|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.

[[Category:Functional analysis]]
[[Category:C*-algebras]]


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