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. Accordingly, we have the following result which guarantees that the abstract and concrete notions of Banach algebra coincide.
Cayley’s theorem for Banach algebras: Every Banach algebra acts faithfully on a Banach space by bounded linear operators such that the norm on agrees with the operator norm.
Proof. Consider the action of on itself by left multiplication; we denote the operator by . This is clearly an injective algebra homomorphism (Cayley’s theorem for algebras). Since we clearly have ; moreover, since we in fact have .
Recall that the spectrum consists of all scalars such that is not invertible. This definition turns out to be very fruitful for Banach algebras in particular due to the following results.
Proposition: Let be an element of a Banach algebra . Then the limit exists.
Proof. Let . Then for any we can find such that , hence . It follows that for we have
hence that . Since this is true for every , we conclude that .
Proposition: Let be such that . Then is invertible.
Proof. The geometric series converges absolutely and is a two-sided inverse for .
Corollary: Let . Then .
Proof. If , then is invertible by the above.
Accordingly, we define the spectral radius
This is the radius of the smallest circle about the origin containing the spectrum of . The above shows that we always have .
For a Banach algebra let , the general linear group of , denote its group of units.
Proposition: is open. The inversion map is continuous (making a topological group).
Proof. By translation it suffices to show that an open neighborhood of the identity in lies in . But this is clear since the open ball of radius, say, about the identity lies in by the above.
Let . Note the identity , which gives
It follows that for fixed we can make arbitrarily small by making sufficiently small.
Corollary: The spectrum is compact.
Proof. The resolvent is defined on the complement of the spectrum (the resolvent set ) and has image in . By the above, is continuous, hence maps open sets to open sets. Since is open in , it follows that is open in 