Boundary (in the theory of uniform algebras)

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.

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