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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201801.png" /> be a [[Uniform algebra|uniform algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201802.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201803.png" /> the algebra of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201804.png" /> (cf. also [[Algebra of functions|Algebra of functions]]). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201805.png" /> is called a Dirichlet algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201806.png" /> is uniformly dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201807.png" />. Dirichlet algebras were introduced by A.M. Gleason [[#References|[a4]]].
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201808.png" /> be a compact subset of the complex plane. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d1201809.png" /> consist of those functions which are analytic on the interior of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018010.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018011.png" /> be the uniform closure in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018012.png" /> of the functions analytic on a neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018013.png" />. T. Gamelin and J. Garnett [[#References|[a3]]] determined exactly when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018014.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018015.png" /> is a Dirichlet algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018016.png" />. The disc algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018017.png" /> is the algebra of all functions which are analytic in the open unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018018.png" /> and continuous in the closed unit disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018019.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018020.png" /> is a typical example of a Dirichlet algebra on the unit circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018021.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018022.png" />, the measure
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018023.png" /></td> </tr></table>
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Let $A$ be a [[Uniform algebra|uniform algebra]] on $X$ and $C ( X )$ the algebra of all continuous functions on $X$ (cf. also [[Algebra of functions|Algebra of functions]]). The algebra $A$ is called a Dirichlet algebra if $A + \overline{A}$ is uniformly dense in $C ( X )$. Dirichlet algebras were introduced by A.M. Gleason [[#References|[a4]]].
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Let $K$ be a compact subset of the complex plane. Let $A ( K )$ consist of those functions which are analytic on the interior of $K$ and let $R ( K )$ be the uniform closure in $C ( K )$ of the functions analytic on a neighbourhood of $K$. T. Gamelin and J. Garnett [[#References|[a3]]] determined exactly when $A ( K )$ or $R ( K )$ is a Dirichlet algebra on $\partial K$. The disc algebra $A ( \mathbf{D} )$ is the algebra of all functions which are analytic in the open unit disc $\mathbf D$ and continuous in the closed unit disc $\overline{\mathbf D }$. The algebra $A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, the measure
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\begin{equation*} \frac { 1 } { 2 \pi } d \theta \end{equation*}
  
 
is the representing measure for the origin, that is,
 
is the representing measure for the origin, that is,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018024.png" /></td> </tr></table>
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\begin{equation*} \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } f ( e ^ { i \theta } ) d \theta = f ( 0 ) \end{equation*}
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018025.png" />. The origin gives a complex [[Homomorphism|homomorphism]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018026.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018027.png" />, the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018028.png" /> is defined as the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018029.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018030.png" /> (cf. also [[Hardy spaces|Hardy spaces]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018031.png" /> be a Dirichlet algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018033.png" /> a non-zero complex homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018035.png" /> is a representing measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018036.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018038.png" /> is unique. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018039.png" />, the abstract Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018040.png" /> is defined as the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018041.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018042.png" />. A lot of theorems for the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018043.png" /> are valid for the abstract Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018044.png" />.
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for $f \in A ( \bf D )$. The origin gives a complex [[Homomorphism|homomorphism]] for $A ( \mathbf{D} )$. For $p \geq 1$, the Hardy space $H ^ { p } ( d \theta / 2 \pi )$ is defined as the closure of $A ( \mathbf{D} )$ in $L ^ { p } ( \partial \mathbf{D} , d \theta / 2 \pi )$ (cf. also [[Hardy spaces|Hardy spaces]]). Let $A$ be a Dirichlet algebra on $X$ and $\phi$ a non-zero complex homomorphism of $A$. If $m$ is a representing measure on $X$ for $\phi$, then $m$ is unique. For $p \geq 1$, the abstract Hardy space $H ^ { p } ( m )$ is defined as the closure of $A$ in $L ^ { p } ( X , m )$. A lot of theorems for the Hardy space $H ^ { p } ( d \theta / 2 \pi )$ are valid for the abstract Hardy space $H ^ { p } ( d m )$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018045.png" /> be a probability measure space (cf. also [[Probability measure|Probability measure]]; [[Measure space|Measure space]]), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018046.png" /> be a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018047.png" /> containing the constants and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018048.png" /> be multiplicative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018049.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018050.png" /> is called a weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018052.png" /> Dirichlet algebra if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018053.png" /> is weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018054.png" /> dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018055.png" />. A Dirichlet algebra is a weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018056.png" /> Dirichlet algebra when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018057.png" /> is a representing measure on it. Weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018058.png" /> Dirichlet algebras were introduced by T. Srinivasan and J. Wang [[#References|[a9]]] as the smallest axiomatic setting on which each one of a lot of important theorems for the Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018059.png" /> are equivalent to the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018060.png" /> is weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018061.png" /> dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018062.png" />.
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Let $( X , \mathcal{B} , m )$ be a probability measure space (cf. also [[Probability measure|Probability measure]]; [[Measure space|Measure space]]), let $A$ be a subalgebra of $L ^ { \infty } ( X , m )$ containing the constants and let $m$ be multiplicative on $A$. The algebra $A$ is called a weak$\square ^ { * }$ Dirichlet algebra if $A + \overline{A}$ is weak$\square ^ { * }$ dense in $L ^ { \infty } ( X , m )$. A Dirichlet algebra is a weak$\square ^ { * }$ Dirichlet algebra when $m$ is a representing measure on it. Weak$\square ^ { * }$ Dirichlet algebras were introduced by T. Srinivasan and J. Wang [[#References|[a9]]] as the smallest axiomatic setting on which each one of a lot of important theorems for the Hardy space $H ^ { p } ( d \theta / 2 \pi )$ are equivalent to the fact that $A + \overline{A}$ is weak$\square ^ { * }$ dense in $L ^ { \infty } ( X , m )$.
  
K. Hoffman and H. Rossi [[#References|[a6]]] gave an example such that even if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018063.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018064.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018065.png" /> is not a weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018066.png" /> Dirichlet algebra. Subsequently, it was shown [[#References|[a6]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018067.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018068.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018069.png" /> is a weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018070.png" /> Dirichlet algebra. W. Arveson [[#References|[a1]]] introduced non-commutative weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018072.png" /> Dirichlet algebras, which are also called subdiagonal algebras.
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K. Hoffman and H. Rossi [[#References|[a6]]] gave an example such that even if $A + \overline{A}$ is dense in $L ^ { 3 } ( X , m )$, $A$ is not a weak$\square ^ { * }$ Dirichlet algebra. Subsequently, it was shown [[#References|[a6]]] that if $A + \overline{A}$ is dense in $L ^ { 4 } ( X , m )$, then $A$ is a weak$\square ^ { * }$ Dirichlet algebra. W. Arveson [[#References|[a1]]] introduced non-commutative weak$\square ^ { * }$ Dirichlet algebras, which are also called subdiagonal algebras.
  
==Examples of (weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018073.png" />) Dirichlet algebras.==
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==Examples of (weak$\square ^ { * }$) Dirichlet algebras.==
  
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018074.png" /> be a compact subset of the complex plane and suppose the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018075.png" /> consists of the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018076.png" /> that can be approximated uniformly on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018077.png" /> by polynomials in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018078.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018079.png" /> is a Dirichlet algebra on the outer boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018080.png" /> [[#References|[a2]]].
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Let $K$ be a compact subset of the complex plane and suppose the algebra $P ( K )$ consists of the functions in $C ( K )$ that can be approximated uniformly on $K$ by polynomials in $z$. Then $P ( K )$ is a Dirichlet algebra on the outer boundary of $K$ [[#References|[a2]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018081.png" /> be the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018082.png" /> endowed with the discrete topology and suppose the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018083.png" /> consists of the functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018084.png" /> whose Fourier coefficients are zero on the [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018085.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018086.png" /> is the compact dual group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018087.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018088.png" /> is a Dirichlet algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018089.png" /> [[#References|[a5]]].
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Let $\mathbf R _ { d}$ be the real line $\mathbf{R}$ endowed with the discrete topology and suppose the algebra $A ( G )$ consists of the functions in $C ( G )$ whose Fourier coefficients are zero on the [[Semi-group|semi-group]] $( \mathbf{R} _ { d } , + )$, where $G$ is the compact dual group of $\mathbf R _ { d}$. Then $A ( G )$ is a Dirichlet algebra on $G$ [[#References|[a5]]].
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018090.png" /> be a fixed compact [[Hausdorff space|Hausdorff space]] upon which the real line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018091.png" /> (with the usual topology) acts as a locally compact [[Transformation group|transformation group]]. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018092.png" /> is called a flow. The translate of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018093.png" /> by a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018094.png" /> is written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018095.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018096.png" /> is called analytic if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018097.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d12018099.png" /> is a boundary function which is bounded and analytic in the upper half-plane. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d120180100.png" /> is an invariant ergodic [[Probability measure|probability measure]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d120180101.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d120180102.png" /> is a weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d120180103.png" /> Dirichlet algebra in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d120180104.png" /> [[#References|[a7]]]. See also [[Hypo-Dirichlet algebra|Hypo-Dirichlet algebra]].
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Let $X$ be a fixed compact [[Hausdorff space|Hausdorff space]] upon which the real line $\mathbf{R}$ (with the usual topology) acts as a locally compact [[Transformation group|transformation group]]. The pair $( X , \mathbf{R} )$ is called a flow. The translate of an $x \in X$ by a $t \in \mathbf{R}$ is written as $x + t$. A $\phi \in C ( X )$ is called analytic if for each $x \in X$ the function $\phi ( x + t )$ of $t$ is a boundary function which is bounded and analytic in the upper half-plane. If $m$ is an invariant ergodic [[Probability measure|probability measure]] on $X$, then $A$ is a weak$\square ^ { * }$ Dirichlet algebra in $L ^ { \infty } ( m )$ [[#References|[a7]]]. See also [[Hypo-Dirichlet algebra|Hypo-Dirichlet algebra]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Arveson,  "Analyticity in operator algebras"  ''Amer. J. Math.'' , '''89'''  (1967)  pp. 578–642</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H. Barbey,  H. König,  "Abstract analytic function theory and Hardy algebras" , ''Lecture Notes Math.'' :  593 , Springer  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  T. Gamelin,  J. Garnett,  "Pointwise bounded approximation and Dirichlet algebras"  ''J. Funct. Anal.'' , '''8'''  (1971)  pp. 360–404</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Gleason,  "Function algebras" , ''Sem. Analytic Functions'' , '''II''' , Inst. Adv. Study Princeton  (1957)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Helson,  "Analyticity on compact Abelian groups" , ''Algebras in Analysis; Proc. Instructional Conf. and NATO Adv. Study Inst., Birmigham, 1973'' , Acad. Press  (1975)  pp. 1–62</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  K. Hoffman,  H. Rossi,  "Function theory from a multiplicative linear functional"  ''Trans. Amer. Math. Soc.'' , '''102'''  (1962)  pp. 507–544</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  P. Muhly,  "Function algebras and flows"  ''Acta Sci. Math.'' , '''35'''  (1973)  pp. 111–121</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  T. Nakazi,  "Hardy spaces and Jensen measures"  ''Trans. Amer. Math. Soc.'' , '''274'''  (1982)  pp. 375–378</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  T. Srinivasan,  J. Wang,  "Weak<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d120/d120180/d120180105.png" />-Dirichlet algebras, Function algebras" , Scott Foresman  (1966)  pp. 216–249</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  J. Wermer,  "Dirichlet algebras"  ''Duke Math. J.'' , '''27'''  (1960)  pp. 373–381</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  W. Arveson,  "Analyticity in operator algebras"  ''Amer. J. Math.'' , '''89'''  (1967)  pp. 578–642</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  H. Barbey,  H. König,  "Abstract analytic function theory and Hardy algebras" , ''Lecture Notes Math.'' :  593 , Springer  (1977)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  T. Gamelin,  J. Garnett,  "Pointwise bounded approximation and Dirichlet algebras"  ''J. Funct. Anal.'' , '''8'''  (1971)  pp. 360–404</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  A. Gleason,  "Function algebras" , ''Sem. Analytic Functions'' , '''II''' , Inst. Adv. Study Princeton  (1957)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  H. Helson,  "Analyticity on compact Abelian groups" , ''Algebras in Analysis; Proc. Instructional Conf. and NATO Adv. Study Inst., Birmigham, 1973'' , Acad. Press  (1975)  pp. 1–62</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  K. Hoffman,  H. Rossi,  "Function theory from a multiplicative linear functional"  ''Trans. Amer. Math. Soc.'' , '''102'''  (1962)  pp. 507–544</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  P. Muhly,  "Function algebras and flows"  ''Acta Sci. Math.'' , '''35'''  (1973)  pp. 111–121</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  T. Nakazi,  "Hardy spaces and Jensen measures"  ''Trans. Amer. Math. Soc.'' , '''274'''  (1982)  pp. 375–378</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  T. Srinivasan,  J. Wang,  "Weak$*$-Dirichlet algebras, Function algebras" , Scott Foresman  (1966)  pp. 216–249</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  J. Wermer,  "Dirichlet algebras"  ''Duke Math. J.'' , '''27'''  (1960)  pp. 373–381</td></tr></table>

Revision as of 16:58, 1 July 2020

Let $A$ be a uniform algebra on $X$ and $C ( X )$ the algebra of all continuous functions on $X$ (cf. also Algebra of functions). The algebra $A$ is called a Dirichlet algebra if $A + \overline{A}$ is uniformly dense in $C ( X )$. Dirichlet algebras were introduced by A.M. Gleason [a4].

Let $K$ be a compact subset of the complex plane. Let $A ( K )$ consist of those functions which are analytic on the interior of $K$ and let $R ( K )$ be the uniform closure in $C ( K )$ of the functions analytic on a neighbourhood of $K$. T. Gamelin and J. Garnett [a3] determined exactly when $A ( K )$ or $R ( K )$ is a Dirichlet algebra on $\partial K$. The disc algebra $A ( \mathbf{D} )$ is the algebra of all functions which are analytic in the open unit disc $\mathbf D$ and continuous in the closed unit disc $\overline{\mathbf D }$. The algebra $A ( \mathbf{D} )$ is a typical example of a Dirichlet algebra on the unit circle $\partial \mathbf{D}$. For $A ( \mathbf{D} )$, the measure

\begin{equation*} \frac { 1 } { 2 \pi } d \theta \end{equation*}

is the representing measure for the origin, that is,

\begin{equation*} \frac { 1 } { 2 \pi } \int _ { 0 } ^ { 2 \pi } f ( e ^ { i \theta } ) d \theta = f ( 0 ) \end{equation*}

for $f \in A ( \bf D )$. The origin gives a complex homomorphism for $A ( \mathbf{D} )$. For $p \geq 1$, the Hardy space $H ^ { p } ( d \theta / 2 \pi )$ is defined as the closure of $A ( \mathbf{D} )$ in $L ^ { p } ( \partial \mathbf{D} , d \theta / 2 \pi )$ (cf. also Hardy spaces). Let $A$ be a Dirichlet algebra on $X$ and $\phi$ a non-zero complex homomorphism of $A$. If $m$ is a representing measure on $X$ for $\phi$, then $m$ is unique. For $p \geq 1$, the abstract Hardy space $H ^ { p } ( m )$ is defined as the closure of $A$ in $L ^ { p } ( X , m )$. A lot of theorems for the Hardy space $H ^ { p } ( d \theta / 2 \pi )$ are valid for the abstract Hardy space $H ^ { p } ( d m )$.

Let $( X , \mathcal{B} , m )$ be a probability measure space (cf. also Probability measure; Measure space), let $A$ be a subalgebra of $L ^ { \infty } ( X , m )$ containing the constants and let $m$ be multiplicative on $A$. The algebra $A$ is called a weak$\square ^ { * }$ Dirichlet algebra if $A + \overline{A}$ is weak$\square ^ { * }$ dense in $L ^ { \infty } ( X , m )$. A Dirichlet algebra is a weak$\square ^ { * }$ Dirichlet algebra when $m$ is a representing measure on it. Weak$\square ^ { * }$ Dirichlet algebras were introduced by T. Srinivasan and J. Wang [a9] as the smallest axiomatic setting on which each one of a lot of important theorems for the Hardy space $H ^ { p } ( d \theta / 2 \pi )$ are equivalent to the fact that $A + \overline{A}$ is weak$\square ^ { * }$ dense in $L ^ { \infty } ( X , m )$.

K. Hoffman and H. Rossi [a6] gave an example such that even if $A + \overline{A}$ is dense in $L ^ { 3 } ( X , m )$, $A$ is not a weak$\square ^ { * }$ Dirichlet algebra. Subsequently, it was shown [a6] that if $A + \overline{A}$ is dense in $L ^ { 4 } ( X , m )$, then $A$ is a weak$\square ^ { * }$ Dirichlet algebra. W. Arveson [a1] introduced non-commutative weak$\square ^ { * }$ Dirichlet algebras, which are also called subdiagonal algebras.

Examples of (weak$\square ^ { * }$) Dirichlet algebras.

Let $K$ be a compact subset of the complex plane and suppose the algebra $P ( K )$ consists of the functions in $C ( K )$ that can be approximated uniformly on $K$ by polynomials in $z$. Then $P ( K )$ is a Dirichlet algebra on the outer boundary of $K$ [a2].

Let $\mathbf R _ { d}$ be the real line $\mathbf{R}$ endowed with the discrete topology and suppose the algebra $A ( G )$ consists of the functions in $C ( G )$ whose Fourier coefficients are zero on the semi-group $( \mathbf{R} _ { d } , + )$, where $G$ is the compact dual group of $\mathbf R _ { d}$. Then $A ( G )$ is a Dirichlet algebra on $G$ [a5].

Let $X$ be a fixed compact Hausdorff space upon which the real line $\mathbf{R}$ (with the usual topology) acts as a locally compact transformation group. The pair $( X , \mathbf{R} )$ is called a flow. The translate of an $x \in X$ by a $t \in \mathbf{R}$ is written as $x + t$. A $\phi \in C ( X )$ is called analytic if for each $x \in X$ the function $\phi ( x + t )$ of $t$ is a boundary function which is bounded and analytic in the upper half-plane. If $m$ is an invariant ergodic probability measure on $X$, then $A$ is a weak$\square ^ { * }$ Dirichlet algebra in $L ^ { \infty } ( m )$ [a7]. See also Hypo-Dirichlet algebra.

References

[a1] W. Arveson, "Analyticity in operator algebras" Amer. J. Math. , 89 (1967) pp. 578–642
[a2] H. Barbey, H. König, "Abstract analytic function theory and Hardy algebras" , Lecture Notes Math. : 593 , Springer (1977)
[a3] T. Gamelin, J. Garnett, "Pointwise bounded approximation and Dirichlet algebras" J. Funct. Anal. , 8 (1971) pp. 360–404
[a4] A. Gleason, "Function algebras" , Sem. Analytic Functions , II , Inst. Adv. Study Princeton (1957)
[a5] H. Helson, "Analyticity on compact Abelian groups" , Algebras in Analysis; Proc. Instructional Conf. and NATO Adv. Study Inst., Birmigham, 1973 , Acad. Press (1975) pp. 1–62
[a6] K. Hoffman, H. Rossi, "Function theory from a multiplicative linear functional" Trans. Amer. Math. Soc. , 102 (1962) pp. 507–544
[a7] P. Muhly, "Function algebras and flows" Acta Sci. Math. , 35 (1973) pp. 111–121
[a8] T. Nakazi, "Hardy spaces and Jensen measures" Trans. Amer. Math. Soc. , 274 (1982) pp. 375–378
[a9] T. Srinivasan, J. Wang, "Weak$*$-Dirichlet algebras, Function algebras" , Scott Foresman (1966) pp. 216–249
[a10] J. Wermer, "Dirichlet algebras" Duke Math. J. , 27 (1960) pp. 373–381
How to Cite This Entry:
Dirichlet algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_algebra&oldid=50275
This article was adapted from an original article by T. Nakazi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article