is.
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 invertible), in which case it generalizes the group algebra, although it is sometimes also given when is a groupoid.
Groupoids naturally arise when some space of interest is quotiented by some equivalence relation or group action. In such situations it often happens that the quotient as a space is very badly behaved, and Alain Connes’ approach to understanding this situation via noncommutative geometry is to replace the quotient by the (noncommutative space corresponding to the) convolution algebra of the corresponding groupoid.
Sub-example. Taking to be the groupoid of objects, all of which are isomorphic via a unique isomorphism, we recover (with norm given by the sum of the absolute values of the entries). This is more or less how Heisenberg discovered matrix mechanics: rather than thinking in terms of vector spaces he thought in terms of transitions between energy levels, which form the morphisms in a groupoid.
The category above is equivalent but not isomorphic to the category with one object and a single morphism. The corresponding algebra is just , which is similarly Morita equivalent but not isomorphic to .
Example. Let be a locally compact group (e.g. ). has a nice left-invariant measure called Haar measure (e.g. Lebesgue measure) which is unique up to scaling; if is compact we can normalize it so that to get normalized Haar measure. The Banach space can also be equipped with a convolution product, which for groups can be written
When is a group with the discrete topology, counting measure is a Haar measure and we recover . Like , typically is not unital (for example when ).
Compact Hausdorff spaces
Theorem: The functors define a contravariant equivalence of categories from the category of compact Hausdorff spaces to a subcategory of the category of -algebras (where is given the Zariski topology).
Banach algebras are an abstraction of closed subalgebras of the algebra of bounded operators on a Banach space in the same way that groups are an abstraction of subgroups of the symmetric group on a set. Accord