Banach algebras abstract the properties of closed algebras of operators on Banach spaces. Many basic properties of such operators have elegant proofs in the framework of Banach algebras, and Banach algebras also naturally appear in areas of mathematics like harmonic analysis, where one writes down Banach algebras generalizing the group algebra to study topological groups.
Today we will develop some of the basic theory of Banach algebras, our goal being to discuss the Gelfand representation of a commutative Banach algebra and the fact that, for commutative C*-algebras, this representation is an isometric isomorphism. This implies in particular a spectral theorem for self-adjoint operators on a Hilbert space.
This material can be found in many sources; I am working from Dales, Aiena, Eschmeier, Laursen and Willis’ Introduction to Banach Algebras, Operators, and Harmonic Analysis.
Below all vector spaces are over , all algebras are unital, and all algebra homomorphisms preserve units unless otherwise stated. In the context of Banach algebras, the last two assumptions are not standard, but in practice non-unital Banach algebras are studied by adjoining units first, so we do not lose much generality.
Definitions and examples
A Banach algebra is a Banach space which is also an associative algebra (with the same addition) such that the multiplication satisfies and . Abstractly, it is a monoid object in . A morphism of Banach algebras is a weak contraction which is also an algebra homomorphism. Note that since we want morphisms to preserve units, any morphism has norm exactly .
(Note also that the zero ring cannot be a Banach algebra with these axioms since and is incompatible with .)
Example. The fundamental example is the algebra bounded linear operators on a Banach space with the operator norm.
Example. Let be a bounded linear operator on a Banach space. Then the closure of the algebra is a commutative Banach algebra. The application of Banach algebra techniques to spectral theory proceeds by studying algebras like this.
Example. Let be a collection of bounded linear operators on a Banach space . Their commutant is an intersection of closed subalgebras of , hence also a Banach algebra. If is thought of as a module over the algebra generated by , then consists of the endomorphisms of this module.
Example. Let be a topological space and consider again the space of bounded continuous functions . This space is a Banach space when equipped with the sup norm, and moreover pointwise multiplication makes it a Banach algebra. If is discrete, this is just . If is compact, this is just .
Example. Let be a locally compact Hausdorff space which is not compact. We denote by the space of continuous functions which vanish at infinity in the sense that for any there exists a compact set such that for ; these are precisely the functions vanishing at the point at infinity in the one-point compactification of . This condition is preserved by uniform limits, so is a Banach space when equipped with the sup norm. When equipped with pointwise products, it is a non-unital Banach algebra (the identity cannot vanish at infinity). Adjoining a unit corresponds to passing to the one-point compactification.
Example. Let be a small category. Recall that the category ring is the free abelian group on the morphisms of , with the product of two morphisms being their composition if defined and zero otherwise (and recall that it is unital if and only if has finitely many objects.) Since it is in particular a free abelian group on a set, we can tensor it with and equip it with the norm; the resulting completion is non-unital if and only if has infinitely many objects. Even if has finitely many objects, in which case the unit of is the sum of the identity morphisms in , that unit satisfies , which is usually not equal to .
Thinking of in terms of functions , the product is the convolution
and so is called the convolution algebra of . This definition is usually only stated in the special case that is a group (that is, a one-object small category with all morphisms 