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Maximal Ideal SpaceEdit

Maximal ideal space is a foundational concept in the study of commutative Banach algebras and their representations. Given a commutative Banach algebra A with identity over the complex numbers, each maximal ideal m ⊂ A corresponds to a multiplicative linear functional φ: A → ℂ with kernel m. Conversely, every nonzero multiplicative linear functional is a character, and its kernel is a maximal ideal. Collectively, the set M(A) of all maximal ideals (or, equivalently, the set of all characters) carries a natural topology that makes it a compact, Hausdorff space known as the maximal ideal space or the spectrum of A. The central tool connecting A to M(A) is the Gelfand transform, which associates to each a ∈ A a continuous function â on M(A) defined by â(φ) = φ(a).

In the special case of unital commutative C*-algebras, this duality is even stronger: the Gelfand transform is a *-isomorphism between the algebra and the algebra of continuous functions on its maximal ideal space, i.e., A ≅ C(M(A)). This viewpoint provides a powerful bridge between algebra, analysis, and topology, turning algebraic questions into questions about functions on a topological space, and vice versa.

Maximal ideal space

Definition and basic properties

  • Let A be a commutative Banach algebra with identity. A maximal ideal m ⊂ A is a proper ideal that is maximal with respect to inclusion. The collection M(A) of all maximal ideals can be identified with the collection of all nonzero multiplicative linear functionals φ: A → ℂ, via m ↔ ker φ.
  • The topology on M(A) is the hull-kernel topology, which coincides with the weak-* topology when M(A) is viewed as a subset of the dual space A′. Under this topology, M(A) is compact and Hausdorff whenever A is unital.
  • The Gelfand transform Ā: A → C(M(A)) is defined by Ā(a)(φ) = φ(a) for φ ∈ M(A). The map a ↦ Ā(a) is a homomorphism of algebras, and it preserves the involution when A is a *-algebra.

Topology and the Gelfand transform

  • The Gelfand topology on M(A) is the weakest topology making all evaluation maps φ → φ(a) continuous for every a ∈ A. When A is unital, M(A) is a compact Hausdorff space in this topology.
  • The Gelfand transform is always continuous, and it separates points of M(A) in the sense that distinct φ give rise to different evaluation functionals on A.
  • For a unital commutative C*-algebra, the Gelfand transform is not just continuous but a *-isomorphism onto C(M(A)). This is the content of the Gelfand–Naimark theorem specialized to abelian algebras.

Examples

  • If A = C(X) for a compact Hausdorff space X, then M(A) is naturally homeomorphic to X, with the maximal ideal corresponding to point evaluations: m_x = {f ∈ A : f(x) = 0}. The Gelfand transform recovers the original function: Ā(a)(x) = a(x).
  • For the disc algebra A(D) (the algebra of functions holomorphic on the open unit disc and continuous on the closed disc), the maximal ideal space M(A(D)) is the closed unit disc. Point evaluations at z ∈ D or on the boundary ∂D yield characters, and the Shilov boundary of A(D) is the unit circle ∂D.
  • The algebra ℓ∞(ℕ) of all bounded sequences under pointwise operations has as its maximal ideal space the Stone–Čech compactification βℕ. This space contains N as a dense subset, but also many “infinite” points corresponding to ultrafilters, reflecting the rich scope of characters in this non-separable setting.
  • In the setting of uniform algebras on a compact space X, the maximal ideal space can be seen as a compactification of X, in which X sits densely via point evaluations, but the full space M(A) may include additional boundary or limit points that encode algebraic constraints on functions in A.

Gelfand theory and applications

  • The representation A ≅ C(M(A)) for abelian C*-algebras allows one to study algebraic properties of A through the topology of M(A) and the regularity of functions on it. This viewpoint is central in spectral theory, harmonic analysis, and approximation theory.
  • Concepts such as the Shilov boundary, peak points, and Choquet boundaries arise when A is a subalgebra of C(X) and one analyzes which points of X or M(A) are distinguished by the algebra’s functions.
  • In function theory, maximal ideal spaces provide a framework to understand extensions, boundary behavior, and the ways in which algebras “see” the underlying geometric or analytic structure.

See also