space of functions , or equivalently a subspace of the product of copies of ; then the weak-* topology is the subspace topology with respect to the product topology on this product.)
Since morphisms have norm , in fact lies in the unit ball of . We now need the following result, which shows that the weak-* topology is very different from the norm topology on .
Theorem (Banach-Alaoglu): let be a normed space. Then the unit ball of is compact in the weak-* topology.
Proof. Thinking of as a subspace of with the product topology, the unit ball consists of linear functionals which send the unit ball of to the unit ball (disk) , hence can be thought of as a subspace of with the product topology. This is compact by Tychonoff’s theorem. The unit ball of is a closed subspace of (since it is defined by the additional requirement that it consists of linear functions, which impose a collection of closed conditions), hence is also compact.
(Note that is compact Hausdorff, so we only need Tychonoff’s theorem for compact Hausdorff spaces here, hence we only need the ultrafilter lemma and not the full strength of the axiom of choice. In fact Banach-Alaoglu is equivalent to the ultrafilter lemma.)
Corollary: For a commutative Banach algebra, is compact Hausdorff in the weak-* topology.
It follows that the homomorphism above, applied to a commutative Banach algebra, gives us its Gelfand representation
Since is compact Hausdorff, the space of continuous functions on it is itself a Banach algebra with the sup norm. We record the following properties of the Gelfand representation to summarize what we have already proven.
- has range the spectrum of .
- Consequently is the spectral radius .
- Consequently the Gelfand representation is a weak contraction (hence a morphism of Banach algebras).
- The kernel of the Gelfand representation is . It is also the set of elements of with spectral radius zero (the quasi-nilpotent elements).
- Consequently the Gelfand representation is faithful if and only if is semiprimitive, if and only if has no nonzero quasi-nilpotents.
(Some sources use “semisimple” to mean semiprimitive. The people who use this convention use “Artinian semisimple” to mean semisimple.)
Commutative semiprimitive Banach algebras therefore have a relatively concrete description as algebras of continuous functions on compact Hausdorff spaces; note, however, that even if is semiprimitive we are not guaranteed that the spectral radius coincides with the norm on . For an important class of commutative Banach algebras which we now introduce, we will be able to show that the Gelfand representation is an isometric isomorphism.
For a Hilbert space, the Banach algebra of bounded linear operators on crucially has extra structure not present in a general Banach algebra given by the adjoint. In abstracting this structure, we are led to the following sequence of definitions.
A *-ring (or involutive ring, or ring with involution) is a ring equipped with an anti-homomorphism which squares to the identity. Explicitly, this is a map satisfying
Equivalently, a *-ring is a small -enriched dagger category, so I am going to call them -rings and use for the involution even though “dagger ring” appears to have some unrelated meaning in mathematics. A morphism of -rings is a ring homomorphism preserving the involution.
Example. Any commutative ring is a -ring with trivial involution.
Example. is a -ring with the involution being complex conjugation.
Example. The quaternions are a -ring with the involution sending to .
Example. Let be a small dagger category. The category ring is a -ring with the involution extending the involution on . In particular, a groupoid is a dagger category with the involution being inverse, so we get an example for any groupoid and in particular any group.
Example. Let be a commutative ring. The ring of matrices over is a -ring with the involution being transpose. (This is more or less a special case of the above applied to the groupoid of equivalent objects after tensoring with .)
A *-algebra over a commutative *-ring is an algebra o