Difference between revisions of "Hypo-Dirichlet algebra"
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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>< | + | <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> |
Revision as of 16:57, 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 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
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