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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]]].

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

\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|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 $( X , \mathcal{B} , m )$ 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 $A + \overline{A}$ 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$\square ^ { * }$) 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 $K$ [[#References|[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|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 $X$ 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====
<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>