A Banach algebra with identity over the field in which for all .

Each maximal ideal of a commutative Banach algebra is the kernel of some continuous multiplicative linear functional on , that is, a homomorphism of into the field of complex numbers. Conversely, every multiplicative linear functional on a commutative Banach algebra is continuous, has norm 1 and its kernel is a maximal ideal in . Let be the set of all multiplicative linear functionals on . An element is invertible if and only if for all . Furthermore, the spectrum consists precisely of the numbers of the form . If a continuous linear functional on has the property that for all , then is multiplicative; this is not true, in general, for an algebra over the field of real numbers.

Examples of maximal ideals in commutative Banach algebras.

Let be the algebra of all continuous functions on a compactum . If is a fixed point of , then the set of all for which is a maximal ideal, and all maximal ideals in have this form. If is a compact set in the complex plane and is the closed subalgebra of consisting of all functions that can be approximated uniformly on by rational functions with poles outside , then the maximal ideals of are obtained in the same way as in the case of . Let be the group algebra of a discrete Abelian group , and suppose that to every element corresponds its Fourier transform . If is a multiplicative linear functional on , then for some in the group of characters of ; therefore the maximal ideals of are in one-to-one correspondence with the elements of . As applied to the group of integers , this last example leads to a proof of the well-known Wiener theorem: If the function has an absolutely convergent Fourier series and does not vanish on , then also has an absolutely convergent Fourier series.

Since a multiplicative linear functional has norm 1, each such a functional belongs to the unit sphere of the dual of . The set of all multiplicative linear functionals on is closed in the weak topology on the dual space. Since the unit ball is compact in the weak topology on the dual space, is also compact in this topology; it is called the maximal ideal space of the algebra and it is denoted by .

If a commutative Banach algebra contains a non-trivial idempotent, that is, an element such that , and , then the maximal ideal space of is disconnected. Conversely, if the maximal ideal space of the algebra is the union of two disjoint closed sets and , then there is an element such that and (Shilov's theorem). In particular, the maximal ideal space of a commutative Banach algebra is connected if and only if this algebra cannot be represented as a direct sum of two non-trivial ideals.

Let be the subgroup of the group of invertible elements of the algebra consisting of the exponentials, that is, of the elements of the form . Then is the connected component of the identity in . For any compactum there is a canonical isomorphism between the groups and , where is the algebra of all continuous functions on (the Brushlinskii–Eilenberg theorem). It turns out that this isomorphism naturally induces an isomorphism between and , where is any commutative Banach algebra whose maximal ideal space is (the Arens–Royden theorem). In some cases the groups with odd have a similar interpretation. The algebra has the following canonical representation in the algebra . The Gel'fand transform of an element is the function on defined by the formula , where is the multiplicative linear functional corresponding to the point . The kernel of the homomorphism is the set of all elements belonging to all maximal ideals, i.e. belonging to the radical of . If is a semi-simple algebra, that is, if , then the homomorphism is an (algebraic) isomorphism of to . Semi-simple commutative Banach algebras are often called function algebras.

The Gel'fand transform is well suited to the study of semi-simple algebras: One of the fundamental results in the theory of commutative Banach algebras is the theorem that a semi-simple algebra can be represented as an algebra of continuous functions on the maximal ideal space. Far less is known about general algebras with a radical in comparison to semi-simple algebras. All ideals of the algebra of complex polynomials of degree are known. This algebra consists of formal polynomials , with the usual multiplication rule, subject to the relation . This algebra is finite-dimensional, all norms on it are equivalent and every ideal of it is closed. The set of those for which for is a closed ideal; there are no other ideals in this algebra. Every algebra with a unique non-trivial ideal is isomorphic to the algebra of polynomials of the first degree. Until now (1987) it is not known whether the same is true for algebras with a unique non-trivial closed ideal.

The natural infinite-dimensional analogues of algebras of polynomials are algebras of power series , with the usual operations and norm , where is a sequence of positive numbers satisfying . If as , then the unique non-trivial homomorphism into the field of complex numbers is given by . Thus, is the unique maximal ideal and this ideal coincides with the radical. The ideals , defined in the same way as in the finite-dimensional case, constitute a countable set of closed ideals. If the sequence is monotone, then this set of ideals contains all closed ideals. In general, an algebra may contain uncountably many distinct closed ideals.

By suitably choosing the sequence in the algebra under consideration (without non-trivial nilpotents), it is possible to define a non-zero derivation, that is, a bounded linear operator such that . There are no non-trivial continuous derivations on a semi-simple algebra, since in any (not necessarily commutative) algebra the identity holds if and commute. In particular, if is continuous, then is a generalized nilpotent.

Any finite-dimensional algebra decomposes into the direct sum of the radical and a semi-simple algebra. In the infinite-dimensional case this assertion ceases to be true in general, even for commutative Banach algebras. In addition, it is necessary to distinguish between the cases of algebraic and strong (topological) decomposability.

It turns out that there are no conditions that can be imposed merely on the radical that will ensure even algebraic decomposability: the radical may be one-dimensional and may annihilate some maximal ideal but it need not be a direct summand, even in the algebraic sense.

On the other hand, if the radical is finite-dimensional and the quotient algebra is an algebra of continuous functions (or an algebra of operators on a Hilbert space), then it is strongly decomposable. If the quotient algebra is an algebra of continuous functions and its annihilator radical (i.e. the square of every element of is zero) has a Banach complement, then is strongly decomposable. Instead of the condition that has a complement one can require that the space of maximal ideals of satisfy the first axiom of countability at every point.

Completely investigated is also the case when the quotient algebra by the radical is the algebra of continuous functions on a totally-disconnected compactum: A necessary and sufficient condition for decomposability is that the idempotents of the original algebra be uniformly bounded.

Let be a bounded domain in and let be the closed subalgebra of consisting of the functions holomorphic on . It is known that under fairly general hypotheses concerning , any maximal ideal of , corresponding to a point , is finitely generated; namely, it is generated by the functions . This statement has the following local converse. Let be a semi-simple commutative Banach algebra with maximal ideal space . If the maximal ideal corresponding to a point is generated by a finite set of elements , then the maximal ideals corresponding to the points in some neighbourhood of are generated by elements of the form ; the mapping is one-to-one in some neighbourhood of and the function is, for any , holomorphic in some fixed neighbourhood of the origin in . Furthermore, in a neighbourhood of a certain natural analytic structure can be introduced.

A set of elements of an algebra is called a system of generators if the smallest closed algebra with identity in that contains is itself. The identity is usually not included in the set of generators. If there is a finite system with the above properties, then is called a finitely-generated algebra. The smallest numbers of elements in a system of generators is called the number of generators of the algebra.

If is a system of generators of an algebra, then the mapping induces a homomorphism of the maximal ideal space of this algebra onto some polynomially-convex compact set in . Each polynomially-convex compact set in is the maximal ideal space of some Banach algebra (for example, the algebra of uniform limits of polynomials on this set).

The maximal ideal space of an algebra with generators satisfies the condition and possesses a number of other properties; for example, for . Hence it follows, in particular, that the number of generators in the algebra , where is the -dimensional unit sphere, is equal to ; a similar result holds for an arbitrary -dimensional compact manifold . For any finite cellular -dimensional polyhedron , the algebra has a system of generators.

Let be an algebra with maximal ideal space . The smallest closet set on which all functions attain their maximum is called the Shilov boundary of . For any commutative Banach algebra with identity this set exists and is unique.

A point belongs to if and only if for any neighbourhood of there is an element for which , but outside . Furthermore, if is an open subset of and if there exist a closed set and an element such that for points , then the intersection is non-empty.

Any multiplicative linear functional is continuous with respect to the norm defined by the spectral radius; moreover, , where is the maximal ideal space. In this inequality, according to the definition of the Shilov boundary, one can replace by ; therefore there exists a positive regular measure on  "representing"  the functional , that is, such that for all the equation holds. In the case of the algebra of functions analytic on the disc, this formula reduces to the classical Poisson formula. Among the representing measures there exists a measure satisfying the Jensen inequality for all .

Let be a commutative Banach algebra with identity and let be a closed subalgebra. The algebra is called a maximal subalgebra of if contains no closed proper subalgebra properly containing . In each sufficiently large algebra there are maximal subalgebras with identity, and even closed subalgebras of codimension 1. In fact, if and are two distinct homomorphisms of the algebra into the field of complex numbers and if , then the kernel of is a closed subalgebra of for which . Similarly, the kernel of a  "point derivation13B10point derivation" , that is, a functional such that , where is a multiplicative functional, is a subalgebra of codimension 1. In the complex case these examples exhaust all subalgebras of codimension 1. In particular, every such subalgebra of the algebra does not separate points of the compactum , since on there are no derivations (not even discontinuous ones). All subalgebras of finite codimension have a similar description.

The algebra of continuous functions on the unit circle that have an analytic continuation inside the unit disc is a maximal subalgebra of the algebra of continuous functions on the unit circle. This statement can be regarded as a generalization of the Stone–Weierstrass approximation theorem, which asserts that a closed subalgebra of , where , containing and the function coincides with . The algebra is a closed subalgebra of ; this subalgebra is maximal.

Let be a irrational number and let be the algebra of all continuous functions on the two-dimensional torus with Fourier coefficients for . This algebra is a maximal subalgebra of the algebra of all continuous functions on the torus. The torus is the Shilov boundary of the algebra , and is a Dirichlet algebra. If the torus is realized as the skeleton of the unit bidisc in , then the maximal ideal space of is identified with the subset of the bidisc described by the equation . The point does not belong to the Shilov boundary, but is a one-point Gleason part. (Two multiplicative functionals and on a uniform algebra belong to the same Gleason part, by definition, if .) The algebra is analytic on the maximal ideal space (in the sense of uniqueness: if for the points of a non-empty open set), even though the real dimension of the maximal ideal space is equal to 3. The algebra of continuous functions on the -dimensional torus having an extension inside the corresponding polydisc is not maximal when , but it is maximal in the class of subalgebras that are invariant with respect to holomorphic automorphisms of the torus.

For references see Banach algebra.