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A subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172901.png" /> of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172902.png" /> of maximal ideals of a [[Commutative Banach algebra|commutative Banach algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172903.png" /> with an identity over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172904.png" /> of complex numbers, on which the moduli of the Gel'fand transforms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172905.png" /> of all elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172906.png" /> attain their maximum (cf. [[Gel'fand representation|Gel'fand representation]]). For example, one can set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172907.png" /> (the trivial boundary). Of interest are non-trivial boundaries with some sort of minimality property. There exists among the closed boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172908.png" /> a minimal one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b0172909.png" />, that is, a closed boundary such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729010.png" /> for every closed boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729011.png" />; it is called the Shilov boundary. The points of the Shilov boundary are characterized by the property that for each neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729012.png" /> of such a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729013.png" /> and every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729014.png" />, there exists an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729015.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729017.png" /> outside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729018.png" />. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729019.png" /> constitute  "the most stable"  part of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729020.png" /> of maximal ideals: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729021.png" /> is a commutative Banach algebra containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729022.png" /> as a subalgebra, then the maximal ideals (multiplicative functionals) corresponding to such points can be extended to maximal ideals (multiplicative functionals) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729023.png" />, whereas for maximal ideals not belonging to the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729024.png" /> such an extension is, in general, not possible. This situation is analogous to the stability of the boundary of the spectrum of a bounded linear operator on a Banach space. A typical example is the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729025.png" /> consisting of the functions continuous on the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729026.png" /> that are analytic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729027.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729028.png" /> can be identified with the closed disc and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729029.png" /> with its topological boundary; maximal ideals corresponding to interior points of the disc cannot be extended to maximal ideals of the algebra of all functions continuous on the boundary in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729030.png" /> is naturally included (according to the maximum principle), while the maximal ideals corresponding to the boundary points can be extended.
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As in the case of algebras of analytic functions there is a local maximum-modulus principle for general commutative Banach algebras: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729031.png" /> is an open subset of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729032.png" />, then
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{{TEX|done}}
  
<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/b/b017/b017290/b01729033.png" /></td> </tr></table>
+
A subset  $  \Gamma $
 +
of the space  $  M _ {A} $
 +
of maximal ideals of a [[Commutative Banach algebra|commutative Banach algebra]]  $  A $
 +
with an identity over the field  $  \mathbf C $
 +
of complex numbers, on which the moduli of the Gel'fand transforms  $  \widehat{a}  $
 +
of all elements  $  a \in A $
 +
attain their maximum (cf. [[Gel'fand representation|Gel'fand representation]]). For example, one can set  $  \Gamma = M _ {A} $(
 +
the trivial boundary). Of interest are non-trivial boundaries with some sort of minimality property. There exists among the closed boundaries  $  \Gamma \subset  M _ {A} $
 +
a minimal one  $  \partial  M _ {A} $,
 +
that is, a closed boundary such that  $  \partial  M _ {A} \subset  \Gamma $
 +
for every closed boundary  $  \Gamma $;  
 +
it is called the ''[[Shilov boundary]]''. The points of the Shilov boundary are characterized by the property that for each neighbourhood  $  V \subset  M _ {A} $
 +
of such a point  $  \xi $
 +
and every  $  \epsilon > 0 $,
 +
there exists an element  $  a \in A $
 +
for which  $  \max  | \widehat{a}  | = 1 $
 +
and  $  | \widehat{a}  | < \epsilon $
 +
outside  $  V $.
 +
The points  $  \xi \in \partial  M _ {A} $
 +
constitute  "the most stable" part of the set  $  M _ {A} $
 +
of maximal ideals: If  $  B $
 +
is a commutative Banach algebra containing  $  A $
 +
as a subalgebra, then the maximal ideals (multiplicative functionals) corresponding to such points can be extended to maximal ideals (multiplicative functionals) of  $  B $,
 +
whereas for maximal ideals not belonging to the boundary  $  \partial  M _ {A} $
 +
such an extension is, in general, not possible. This situation is analogous to the stability of the boundary of the spectrum of a bounded linear operator on a Banach space. A typical example is the algebra  $  A $
 +
consisting of the functions continuous on the disc  $  | \lambda | \leq  1 $
 +
that are analytic for  $  | \lambda | < 1 $.
 +
In this case  $  M _ {A} $
 +
can be identified with the closed disc and  $  \partial  M _ {A} $
 +
with its topological boundary; maximal ideals corresponding to interior points of the disc cannot be extended to maximal ideals of the algebra of all functions continuous on the boundary in which  $  A $
 +
is naturally included (according to the maximum principle), while the maximal ideals corresponding to the boundary points can be extended.
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729035.png" /> is the closure and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729036.png" /> is the topological boundary of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729037.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729038.png" />. Roughly speaking, this means that a local maximum point is necessarily a global maximum point (possibly of another element).
+
As in the case of algebras of analytic functions there is a local maximum-modulus principle for general commutative Banach algebras: If  $  V $
 +
is an open subset of the space  $  M _ {A} $,
 +
then
  
The notion of a boundary is used in the study of uniform algebras, that is, closed subalgebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729039.png" /> of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729040.png" /> of all continuous functions on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729041.png" /> that separate points and contain the constants. In this situation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729042.png" />. There exists for each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729043.png" /> a representing measure, concentrated on the Shilov boundary, that is, a probability measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729044.png" /> such that
+
$$
 +
\max \{ {| \widehat{a}  ( \xi ) | } : {\xi \in \overline{V}\; } \}
 +
= \
 +
\max \{ {| \widehat{a}  ( \xi ) | } : {
 +
\xi \in ( \partial  M _ {A} \cap V) \cup \partial  V } \}
 +
$$
  
<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/b/b017/b017290/b01729045.png" /></td> </tr></table>
+
for all  $  a \in A $,
 +
where  $  \overline{V}\; $
 +
is the closure and  $  \partial  V $
 +
is the topological boundary of  $  V $
 +
in  $  M _ {A} $.  
 +
Roughly speaking, this means that a local maximum point is necessarily a global maximum point (possibly of another element).
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729046.png" /> (this holds for arbitrary commutative Banach algebras). In the simplest case (described above) of the disc and the algebra of analytic functions, the latter formula reduces to the [[Poisson formula|Poisson formula]]. The representing measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729047.png" /> is in general not unique. For points belonging to the same Gleason part (see [[Algebra of functions|Algebra of functions]]), the measures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729048.png" /> can be chosen absolutely continuous with respect to one another, which, under certain extra conditions of uniqueness type for the representing measures, makes it possible to equip the Gleason parts of the space of maximal ideals with a one-dimensional analytic structure compatible with the algebra. Every point of the Shilov boundary of a uniform algebra constitutes a one-point Gleason part; however, the converse is in general false.
+
The notion of a boundary is used in the study of uniform algebras, that is, closed subalgebras  $  A $
 +
of the algebra  $  C (X) $
 +
of all continuous functions on a compactum  $  X $
 +
that separate points and contain the constants. In this situation  $  \partial  M _ {A} \subset  X \subset  M _ {A} $.  
 +
There exists for each point  $  \xi \in M _ {A} $
 +
a representing measure, concentrated on the Shilov boundary, that is, a probability measure  $  \mu $
 +
such that
  
The equality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729049.png" /> for uniform algebras is a simple necessary condition for the coincidence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729050.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729051.png" />. Without extra hypotheses this is not a sufficient condition for the above coincidence even in the case of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729052.png" /> of uniform limits of rational functions on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729053.png" /> in the plane.
+
$$
 +
\widehat{a}  ( \xi )  = \
 +
\int\limits _ {\partial  M _ {A} }
 +
\widehat{a}  d \mu
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729054.png" /> be a metrizable compactum and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729055.png" /> be a uniform algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729056.png" />. Then there exists a minimal boundary among all the boundaries: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729057.png" />. The closure of the minimal boundary is the same as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729058.png" />. However, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729059.png" /> is, in general, not closed; an example is the subalgebra of all analytic functions inside the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729060.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729061.png" />. The boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729062.png" /> is the same as the set of peak points with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729063.png" />, that is, points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729064.png" /> for which there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729065.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729066.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729067.png" />. On the other hand, a condition for a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729068.png" /> to belong to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729069.png" /> that is formally considerably weaker is known. Namely, if the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729070.png" /> is such that for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729072.png" /> there exists for each neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729074.png" /> an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729075.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729076.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729078.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729079.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729080.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729081.png" />. The abstract Poisson formula has been strengthened in the following way: There exists for any point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729082.png" /> a representing measure concentrated on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729083.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729084.png" /> being, moreover, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729085.png" />-set). In this form it has been successfully applied in certain problems of approximation theory. The points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729086.png" /> are characterized by the property that for these points, the measure is unique and is the same as the Dirac <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729087.png" />-measure, that is, the minimal boundary is a special case of the Choquet boundary.
+
for all  $  a \in A $(
 +
this holds for arbitrary commutative Banach algebras). In the simplest case (described above) of the disc and the algebra of analytic functions, the latter formula reduces to the [[Poisson formula|Poisson formula]]. The representing measure  $  \mu $
 +
is in general not unique. For points belonging to the same Gleason part (see [[Algebra of functions|Algebra of functions]]), the measures  $  \mu $
 +
can be chosen absolutely continuous with respect to one another, which, under certain extra conditions of uniqueness type for the representing measures, makes it possible to equip the Gleason parts of the space of maximal ideals with a one-dimensional analytic structure compatible with the algebra. Every point of the Shilov boundary of a uniform algebra constitutes a one-point Gleason part; however, the converse is in general false.
  
For the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729088.png" /> of uniform limits of rational functions on a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729089.png" /> in the plane, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729091.png" /> coincide if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729092.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729093.png" /> coincide. This is not true for arbitrary uniform algebras: There exists a uniform algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729094.png" /> distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729095.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729096.png" /> is metrizable and for which each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729097.png" /> is a peak point (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729098.png" />). There also exists a uniform algebra for which all the Gleason parts are trivial (singletons), but for which even the Shilov boundary is a proper part of the space of maximal ideals.
+
The equality  $  \partial  M _ {A} = X = M _ {A} $
 +
for uniform algebras is a simple necessary condition for the coincidence of  $  A $
 +
with  $  C (X) $.  
 +
Without extra hypotheses this is not a sufficient condition for the above coincidence even in the case of algebras  $  A = R (X) $
 +
of uniform limits of rational functions on a compactum  $  X $
 +
in the plane.
  
One of the algebraic generalizations of the notion of the Shilov boundary is the following. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b01729099.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290100.png" /> be the closed ideal generated by a set of elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290102.png" />. The space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290103.png" /> can be naturally identified with the set of common zeros of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290104.png" />. The closure of the union of the boundaries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290105.png" /> over all ideals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290106.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290107.png" /> elements is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290108.png" />. For example, for the algebra of continuous functions in the polydisc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290110.png" /> that are analytic in its interior, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290111.png" /> is the skeleton <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290113.png" /> is the topological boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290115.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290117.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290118.png" />. These generalizations are useful when proving theorems on multi-dimensional analytic structures in spaces of maximal ideals.
+
Let  $  X $
 +
be a metrizable compactum and let  $  A $
 +
be a uniform algebra on  $  X $.
 +
Then there exists a minimal boundary among all the boundaries:  $  \partial  _ {0} M _ {A} $.
 +
The closure of the minimal boundary is the same as  $  \partial  M _ {A} $.
 +
However,  $  \partial  _ {0} M _ {A} $
 +
is, in general, not closed; an example is the subalgebra of all analytic functions inside the disc  $  | \lambda | \leq  1 $
 +
for which  $  f (0) = f (1) $.
 +
The boundary $  \partial  _ {0} M _ {A} $
 +
is the same as the set of peak points with respect to  $  A $,
 +
that is, points  $  \xi \in M _ {A} $
 +
for which there exists an  $  a \in A $
 +
with  $  | \widehat{a}  ( \xi ) | > | \widehat{a}  ( \eta ) | $
 +
for all  $  \eta \neq \xi $.
 +
On the other hand, a condition for a point  $  \xi $
 +
to belong to  $  \partial  _ {0} M _ {A} $
 +
that is formally considerably weaker is known. Namely, if the point  $  \xi \in M _ {A} $
 +
is such that for some  $  0 < c < 1 $
 +
and  $  d \geq  1 $
 +
there exists for each neighbourhood  $  V $
 +
of  $  \xi $
 +
an element  $  a $
 +
in  $  A $
 +
for which  $  \widehat{a}  ( \xi ) = 1 $,
 +
$  \max  | \widehat{a}  | = d $
 +
and  $  | \widehat{a}  ( \eta ) | \leq  c $
 +
for  $  \eta \notin V $,
 +
then  $  \xi \in \partial  _ {0} M _ {A} $.
 +
The abstract Poisson formula has been strengthened in the following way: There exists for any point  $  \xi \in M _ {A} $
 +
a representing measure concentrated on  $  \partial  _ {0} M _ {A} $(
 +
$  \partial  _ {0} M _ {A} $
 +
being, moreover, a  $  G _  \delta  $-
 +
set). In this form it has been successfully applied in certain problems of approximation theory. The points  $  \xi \in \partial  _ {0} M _ {A} $
 +
are characterized by the property that for these points, the measure is unique and is the same as the Dirac  $  \delta $-
 +
measure, that is, the minimal boundary is a special case of the Choquet boundary.
 +
 
 +
For the algebra  $  A = R (X) $
 +
of uniform limits of rational functions on a compactum  $  X $
 +
in the plane,  $  A $
 +
and  $  C (X) $
 +
coincide if and only if  $  X = M _ {A} $
 +
and  $  \partial  _ {0} M _ {A} $
 +
coincide. This is not true for arbitrary uniform algebras: There exists a uniform algebra  $  A $
 +
distinct from  $  C (M _ {A} ) $
 +
for which  $  M _ {A} $
 +
is metrizable and for which each point  $  \xi \in M _ {A} $
 +
is a peak point (that is,  $  \partial  _ {0} M _ {A} = M _ {A} $).  
 +
There also exists a uniform algebra for which all the Gleason parts are trivial (singletons), but for which even the Shilov boundary is a proper part of the space of maximal ideals.
 +
 
 +
One of the algebraic generalizations of the notion of the Shilov boundary is the following. Let  $  \partial  _ {1} M _ {A} = M _ {A} $
 +
and let  $  I $
 +
be the closed ideal generated by a set of elements $  a _ {1} \dots a _ {n - 1 }  \in A $,  
 +
$  n \geq  2 $.  
 +
The space $  M _ {A/I} $
 +
can be naturally identified with the set of common zeros of the functions $  \widehat{a}  _ {1} \dots \widehat{a}  _ {n - 1 }  $.  
 +
The closure of the union of the boundaries $  \partial  M _ {A/I} $
 +
over all ideals $  I $
 +
generated by $  n - 1 $
 +
elements is denoted by $  \partial  _ {n} M _ {A} $.  
 +
For example, for the algebra of continuous functions in the polydisc $  | \lambda _ {1} | \leq  1 $,  
 +
$  | \lambda _ {2} | \leq  1 $
 +
that are analytic in its interior, $  \partial  _ {1} M _ {A} $
 +
is the skeleton $  | \lambda _ {1} | = | \lambda _ {2} | = 1 $,  
 +
$  \partial  _ {2} M _ {A} $
 +
is the topological boundary $  | \lambda _ {1} | = 1 $,  
 +
$  | \lambda _ {2} | \leq  1 $;  
 +
$  | \lambda _ {1} | \leq  1 $,  
 +
$  | \lambda _ {2} | = 1 $,  
 +
and $  \partial  _ {3} M _ {A} = M _ {A} $.  
 +
These generalizations are useful when proving theorems on multi-dimensional analytic structures in spaces of maximal ideals.
  
 
Notions of boundaries (or functional boundaries) similar to the ones described above are encountered in the theory of analytic functions (the Bergman boundary), probability theory (the Martin boundary) and in a number of other branches of mathematics. In this connection the initial set is not necessarily assumed to be an algebra or a ring.
 
Notions of boundaries (or functional boundaries) similar to the ones described above are encountered in the theory of analytic functions (the Bergman boundary), probability theory (the Martin boundary) and in a number of other branches of mathematics. In this connection the initial set is not necessarily assumed to be an algebra or a ring.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R. Basener,  "A generalized Shilov boundary and analytic structure"  ''Proc. Amer. Math. Soc.'' , '''47''' :  1  (1975)  pp. 98–104</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Browder,  "Introduction to function algebras" , Benjamin  (1969)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras" , Prentice-Hall  (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.M. [I.M. Gel'fand] Gelfand,  D.A. [D.A. Raikov] Raikov,  G.E. [G.E. Shilov] Schilow,  "Kommutative Normierte Ringe" , Deutsch. Verlag Wissenschaft.  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Gonchar,  "The minimal boundary of the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017290/b017290119.png" />"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''27''' :  4  (1963)  pp. 949–955  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top">  R. Basener,  "A generalized Shilov boundary and analytic structure"  ''Proc. Amer. Math. Soc.'' , '''47''' :  1  (1975)  pp. 98–104</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Browder,  "Introduction to function algebras" , Benjamin  (1969)</TD></TR>
 
+
<TR><TD valign="top">[3]</TD> <TD valign="top">  T.W. Gamelin,  "Uniform algebras" , Prentice-Hall  (1969)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.M. [I.M. Gel'fand] Gelfand,  D.A. [D.A. Raikov] Raikov,  G.E. [G.E. Shilov] Schilow,  "Kommutative Normierte Ringe" , Deutsch. Verlag Wissenschaft.  (1964)  (Translated from Russian)</TD></TR>
 +
<TR><TD valign="top">[5]</TD> <TD valign="top">  A.A. Gonchar,  "The minimal boundary of the algebra $A(E)$"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''27''' :  4  (1963)  pp. 949–955  (In Russian)</TD></TR>
 +
<TR><TD valign="top">[6]</TD> <TD valign="top">  R.R. Phelps,  "Lectures on Choquet's theorem" , v. Nostrand  (1966)</TD></TR>
 +
</table>
  
 
====Comments====
 
====Comments====
 
Much of this article, in particular the definition of boundary, is relevant and valid for general commutative Banach algebras.
 
Much of this article, in particular the definition of boundary, is relevant and valid for general commutative Banach algebras.

Latest revision as of 08:37, 26 March 2023


A subset $ \Gamma $ of the space $ M _ {A} $ of maximal ideals of a commutative Banach algebra $ A $ with an identity over the field $ \mathbf C $ of complex numbers, on which the moduli of the Gel'fand transforms $ \widehat{a} $ of all elements $ a \in A $ attain their maximum (cf. Gel'fand representation). For example, one can set $ \Gamma = M _ {A} $( the trivial boundary). Of interest are non-trivial boundaries with some sort of minimality property. There exists among the closed boundaries $ \Gamma \subset M _ {A} $ a minimal one $ \partial M _ {A} $, that is, a closed boundary such that $ \partial M _ {A} \subset \Gamma $ for every closed boundary $ \Gamma $; it is called the Shilov boundary. The points of the Shilov boundary are characterized by the property that for each neighbourhood $ V \subset M _ {A} $ of such a point $ \xi $ and every $ \epsilon > 0 $, there exists an element $ a \in A $ for which $ \max | \widehat{a} | = 1 $ and $ | \widehat{a} | < \epsilon $ outside $ V $. The points $ \xi \in \partial M _ {A} $ constitute "the most stable" part of the set $ M _ {A} $ of maximal ideals: If $ B $ is a commutative Banach algebra containing $ A $ as a subalgebra, then the maximal ideals (multiplicative functionals) corresponding to such points can be extended to maximal ideals (multiplicative functionals) of $ B $, whereas for maximal ideals not belonging to the boundary $ \partial M _ {A} $ such an extension is, in general, not possible. This situation is analogous to the stability of the boundary of the spectrum of a bounded linear operator on a Banach space. A typical example is the algebra $ A $ consisting of the functions continuous on the disc $ | \lambda | \leq 1 $ that are analytic for $ | \lambda | < 1 $. In this case $ M _ {A} $ can be identified with the closed disc and $ \partial M _ {A} $ with its topological boundary; maximal ideals corresponding to interior points of the disc cannot be extended to maximal ideals of the algebra of all functions continuous on the boundary in which $ A $ is naturally included (according to the maximum principle), while the maximal ideals corresponding to the boundary points can be extended.

As in the case of algebras of analytic functions there is a local maximum-modulus principle for general commutative Banach algebras: If $ V $ is an open subset of the space $ M _ {A} $, then

$$ \max \{ {| \widehat{a} ( \xi ) | } : {\xi \in \overline{V}\; } \} = \ \max \{ {| \widehat{a} ( \xi ) | } : { \xi \in ( \partial M _ {A} \cap V) \cup \partial V } \} $$

for all $ a \in A $, where $ \overline{V}\; $ is the closure and $ \partial V $ is the topological boundary of $ V $ in $ M _ {A} $. Roughly speaking, this means that a local maximum point is necessarily a global maximum point (possibly of another element).

The notion of a boundary is used in the study of uniform algebras, that is, closed subalgebras $ A $ of the algebra $ C (X) $ of all continuous functions on a compactum $ X $ that separate points and contain the constants. In this situation $ \partial M _ {A} \subset X \subset M _ {A} $. There exists for each point $ \xi \in M _ {A} $ a representing measure, concentrated on the Shilov boundary, that is, a probability measure $ \mu $ such that

$$ \widehat{a} ( \xi ) = \ \int\limits _ {\partial M _ {A} } \widehat{a} d \mu $$

for all $ a \in A $( this holds for arbitrary commutative Banach algebras). In the simplest case (described above) of the disc and the algebra of analytic functions, the latter formula reduces to the Poisson formula. The representing measure $ \mu $ is in general not unique. For points belonging to the same Gleason part (see Algebra of functions), the measures $ \mu $ can be chosen absolutely continuous with respect to one another, which, under certain extra conditions of uniqueness type for the representing measures, makes it possible to equip the Gleason parts of the space of maximal ideals with a one-dimensional analytic structure compatible with the algebra. Every point of the Shilov boundary of a uniform algebra constitutes a one-point Gleason part; however, the converse is in general false.

The equality $ \partial M _ {A} = X = M _ {A} $ for uniform algebras is a simple necessary condition for the coincidence of $ A $ with $ C (X) $. Without extra hypotheses this is not a sufficient condition for the above coincidence even in the case of algebras $ A = R (X) $ of uniform limits of rational functions on a compactum $ X $ in the plane.

Let $ X $ be a metrizable compactum and let $ A $ be a uniform algebra on $ X $. Then there exists a minimal boundary among all the boundaries: $ \partial _ {0} M _ {A} $. The closure of the minimal boundary is the same as $ \partial M _ {A} $. However, $ \partial _ {0} M _ {A} $ is, in general, not closed; an example is the subalgebra of all analytic functions inside the disc $ | \lambda | \leq 1 $ for which $ f (0) = f (1) $. The boundary $ \partial _ {0} M _ {A} $ is the same as the set of peak points with respect to $ A $, that is, points $ \xi \in M _ {A} $ for which there exists an $ a \in A $ with $ | \widehat{a} ( \xi ) | > | \widehat{a} ( \eta ) | $ for all $ \eta \neq \xi $. On the other hand, a condition for a point $ \xi $ to belong to $ \partial _ {0} M _ {A} $ that is formally considerably weaker is known. Namely, if the point $ \xi \in M _ {A} $ is such that for some $ 0 < c < 1 $ and $ d \geq 1 $ there exists for each neighbourhood $ V $ of $ \xi $ an element $ a $ in $ A $ for which $ \widehat{a} ( \xi ) = 1 $, $ \max | \widehat{a} | = d $ and $ | \widehat{a} ( \eta ) | \leq c $ for $ \eta \notin V $, then $ \xi \in \partial _ {0} M _ {A} $. The abstract Poisson formula has been strengthened in the following way: There exists for any point $ \xi \in M _ {A} $ a representing measure concentrated on $ \partial _ {0} M _ {A} $( $ \partial _ {0} M _ {A} $ being, moreover, a $ G _ \delta $- set). In this form it has been successfully applied in certain problems of approximation theory. The points $ \xi \in \partial _ {0} M _ {A} $ are characterized by the property that for these points, the measure is unique and is the same as the Dirac $ \delta $- measure, that is, the minimal boundary is a special case of the Choquet boundary.

For the algebra $ A = R (X) $ of uniform limits of rational functions on a compactum $ X $ in the plane, $ A $ and $ C (X) $ coincide if and only if $ X = M _ {A} $ and $ \partial _ {0} M _ {A} $ coincide. This is not true for arbitrary uniform algebras: There exists a uniform algebra $ A $ distinct from $ C (M _ {A} ) $ for which $ M _ {A} $ is metrizable and for which each point $ \xi \in M _ {A} $ is a peak point (that is, $ \partial _ {0} M _ {A} = M _ {A} $). There also exists a uniform algebra for which all the Gleason parts are trivial (singletons), but for which even the Shilov boundary is a proper part of the space of maximal ideals.

One of the algebraic generalizations of the notion of the Shilov boundary is the following. Let $ \partial _ {1} M _ {A} = M _ {A} $ and let $ I $ be the closed ideal generated by a set of elements $ a _ {1} \dots a _ {n - 1 } \in A $, $ n \geq 2 $. The space $ M _ {A/I} $ can be naturally identified with the set of common zeros of the functions $ \widehat{a} _ {1} \dots \widehat{a} _ {n - 1 } $. The closure of the union of the boundaries $ \partial M _ {A/I} $ over all ideals $ I $ generated by $ n - 1 $ elements is denoted by $ \partial _ {n} M _ {A} $. For example, for the algebra of continuous functions in the polydisc $ | \lambda _ {1} | \leq 1 $, $ | \lambda _ {2} | \leq 1 $ that are analytic in its interior, $ \partial _ {1} M _ {A} $ is the skeleton $ | \lambda _ {1} | = | \lambda _ {2} | = 1 $, $ \partial _ {2} M _ {A} $ is the topological boundary $ | \lambda _ {1} | = 1 $, $ | \lambda _ {2} | \leq 1 $; $ | \lambda _ {1} | \leq 1 $, $ | \lambda _ {2} | = 1 $, and $ \partial _ {3} M _ {A} = M _ {A} $. These generalizations are useful when proving theorems on multi-dimensional analytic structures in spaces of maximal ideals.

Notions of boundaries (or functional boundaries) similar to the ones described above are encountered in the theory of analytic functions (the Bergman boundary), probability theory (the Martin boundary) and in a number of other branches of mathematics. In this connection the initial set is not necessarily assumed to be an algebra or a ring.

References

[1] R. Basener, "A generalized Shilov boundary and analytic structure" Proc. Amer. Math. Soc. , 47 : 1 (1975) pp. 98–104
[2] A. Browder, "Introduction to function algebras" , Benjamin (1969)
[3] T.W. Gamelin, "Uniform algebras" , Prentice-Hall (1969)
[4] I.M. [I.M. Gel'fand] Gelfand, D.A. [D.A. Raikov] Raikov, G.E. [G.E. Shilov] Schilow, "Kommutative Normierte Ringe" , Deutsch. Verlag Wissenschaft. (1964) (Translated from Russian)
[5] A.A. Gonchar, "The minimal boundary of the algebra $A(E)$" Izv. Akad. Nauk SSSR Ser. Mat. , 27 : 4 (1963) pp. 949–955 (In Russian)
[6] R.R. Phelps, "Lectures on Choquet's theorem" , v. Nostrand (1966)

Comments

Much of this article, in particular the definition of boundary, is relevant and valid for general commutative Banach algebras.

How to Cite This Entry:
Boundary (in the theory of uniform algebras). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_(in_the_theory_of_uniform_algebras)&oldid=16227
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article