Dirichlet algebra
{{Short description|Algebraic structure}}
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 {{harvs|first=Andrew|last=Gleason|authorlink=Andrew Gleason|year=1957|txt}}.
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
- {{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}}.
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