View source for Boundary (in the theory of uniform algebras)

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.

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

<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>

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).

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

<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 <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 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.

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 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.

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.

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====
<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>



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