Dirichlet algebra

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