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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201501.png" /> be a [[Uniform algebra|uniform algebra]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201502.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201503.png" /> the algebra of all continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201504.png" /> (cf. also [[Algebra of functions|Algebra of functions]]). The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201505.png" /> is called a hypo-Dirichlet algebra if the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201506.png" /> has finite [[Codimension|codimension]] in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201507.png" />, and the linear span of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201508.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h1201509.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015010.png" /> is the family of invertible elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015011.png" />. Hypo-Dirichlet algebras were introduced by J. Wermer [[#References|[a4]]].
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015012.png" /> be the boundary of a compact subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015013.png" /> in the complex plane whose complement has only finitely many components. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015014.png" /> be the algebra of all functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015015.png" /> that can be uniformly approximated by rational functions with poles off <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015016.png" /> (cf. also [[Padé approximation|Padé approximation]]; [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]]). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015017.png" /> is a hypo-Dirichlet algebra [[#References|[a3]]].
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015018.png" /> be a hypo-Dirichlet algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015020.png" /> a non-zero complex [[Homomorphism|homomorphism]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015021.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015022.png" /> is a representing measure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015024.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015026.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015027.png" /> is unique. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015028.png" />, the abstract Hardy space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015029.png" /> is defined as the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015030.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015031.png" /> (cf. also [[Hardy spaces|Hardy spaces]]). Then a lot of theorems for the concrete Hardy space defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015032.png" /> are valid for abstract Hardy spaces [[#References|[a2]]]. Using such a theory, J. Wermer [[#References|[a4]]] showed that if the Gleason part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015033.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015034.png" /> is non-trivial (cf. also [[Algebra of functions|Algebra of functions]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015035.png" /> has an analytic structure.
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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 hypo-Dirichlet algebra if the closure of $A + \overline{A}$ has finite [[Codimension|codimension]] in $C ( X )$, and the [[linear span]] of $\operatorname { log } | A ^ { - 1 } |$ is dense in $\operatorname { Re } C ( X )$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [[#References|[a4]]].
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Let $X$ be the boundary of a compact subset $Y$ in the complex plane whose complement has only finitely many components. Let $R ( X )$ be the algebra of all functions on $X$ that can be uniformly approximated by rational functions with poles off $Y$ (cf. also [[Padé approximation|Padé approximation]]; [[Approximation of functions of a complex variable|Approximation of functions of a complex variable]]). Then $R ( X )$ is a hypo-Dirichlet algebra [[#References|[a3]]].
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Let $A$ be a hypo-Dirichlet algebra on $X$ and $\phi$ a non-zero complex [[Homomorphism|homomorphism]] of $A$. If $m$ is a representing measure on $X$ such that $\operatorname { log } | \phi ( h ) | = \int \operatorname { log } | h | dm$ for $h$ in $A ^ { - 1 }$, then $m$ is unique. For $p \geq 1$, the abstract Hardy space $H ^ { p } ( m )$ is defined as the closure of $A$ in $L ^ { p } ( X , m )$ (cf. also [[Hardy spaces|Hardy spaces]]). Then a lot of theorems for the concrete Hardy space defined by $R ( X )$ are valid for abstract Hardy spaces [[#References|[a2]]]. Using such a theory, J. Wermer [[#References|[a4]]] showed that if the Gleason part $G ( \phi )$ of $\phi$ is non-trivial (cf. also [[Algebra of functions|Algebra of functions]]), then $G ( \phi )$ has an analytic structure.
  
 
See also [[Dirichlet algebra|Dirichlet algebra]].
 
See also [[Dirichlet algebra|Dirichlet algebra]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P. Ahern,  D. Sarason,  "On some hypodirichlet algebras of analytic functions"  ''Amer. J. Math.'' , '''89'''  (1967)  pp. 932–941</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Ahern,  D. Sarason,  "The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h120/h120150/h12015036.png" /> spaces of a class of function algebras"  ''Acta Math.'' , '''117'''  (1967)  pp. 123–163</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  J. Wermer,  "Analytic disks in maximal ideal spaces"  ''Amer. J. Math.'' , '''86'''  (1964)  pp. 161–170</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  P. Ahern,  D. Sarason,  "On some hypodirichlet algebras of analytic functions"  ''Amer. J. Math.'' , '''89'''  (1967)  pp. 932–941</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  P. Ahern,  D. Sarason,  "The $H ^ { p }$ spaces of a class of function algebras"  ''Acta Math.'' , '''117'''  (1967)  pp. 123–163</td></tr><tr><td valign="top">[a3]</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">[a4]</td> <td valign="top">  J. Wermer,  "Analytic disks in maximal ideal spaces"  ''Amer. J. Math.'' , '''86'''  (1964)  pp. 161–170</td></tr></table>

Latest revision as of 19:56, 27 February 2021

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 hypo-Dirichlet algebra if the closure of $A + \overline{A}$ has finite codimension in $C ( X )$, and the linear span of $\operatorname { log } | A ^ { - 1 } |$ is dense in $\operatorname { Re } C ( X )$, where $A ^ { - 1 }$ is the family of invertible elements of $A$. Hypo-Dirichlet algebras were introduced by J. Wermer [a4].

Let $X$ be the boundary of a compact subset $Y$ in the complex plane whose complement has only finitely many components. Let $R ( X )$ be the algebra of all functions on $X$ that can be uniformly approximated by rational functions with poles off $Y$ (cf. also Padé approximation; Approximation of functions of a complex variable). Then $R ( X )$ is a hypo-Dirichlet algebra [a3].

Let $A$ be a hypo-Dirichlet algebra on $X$ and $\phi$ a non-zero complex homomorphism of $A$. If $m$ is a representing measure on $X$ such that $\operatorname { log } | \phi ( h ) | = \int \operatorname { log } | h | dm$ for $h$ in $A ^ { - 1 }$, then $m$ is unique. For $p \geq 1$, the abstract Hardy space $H ^ { p } ( m )$ is defined as the closure of $A$ in $L ^ { p } ( X , m )$ (cf. also Hardy spaces). Then a lot of theorems for the concrete Hardy space defined by $R ( X )$ are valid for abstract Hardy spaces [a2]. Using such a theory, J. Wermer [a4] showed that if the Gleason part $G ( \phi )$ of $\phi$ is non-trivial (cf. also Algebra of functions), then $G ( \phi )$ has an analytic structure.

See also Dirichlet algebra.

References

[a1] P. Ahern, D. Sarason, "On some hypodirichlet algebras of analytic functions" Amer. J. Math. , 89 (1967) pp. 932–941
[a2] P. Ahern, D. Sarason, "The $H ^ { p }$ spaces of a class of function algebras" Acta Math. , 117 (1967) pp. 123–163
[a3] H. Barbey, H.König, "Abstract analytic function theory and Hardy algebras" , Lecture Notes Math. : 593 , Springer (1977)
[a4] J. Wermer, "Analytic disks in maximal ideal spaces" Amer. J. Math. , 86 (1964) pp. 161–170
How to Cite This Entry:
Hypo-Dirichlet algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hypo-Dirichlet_algebra&oldid=18131
This article was adapted from an original article by T. Nakazi (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article