o 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 over in the ordinary sense which is also equipped with an involution making it a *-ring compatible with the one on . Equivalently, it is a *-ring together with a morphism of *-rings. Again, I will call these -algebras. A morphism of -algebras is an -linear morphism of -rings.
Example. Let be a -ring which is also an algebra over (with the involution restricted to being trivial). Then the tensor product is a complex -algebra (that is, a -algebra over as a -ring) with the involution being complex conjugation on and the existing involution on .
Letting be the category ring of a dagger category we get the complex -algebras .
Example. The algebra is a complex -algebra with the involution being adjoint, as is any -closed subalgebra.
We will primarily be interested in complex -algebras, which are -algebras over regarded as a -ring using complex conjugation. In the commutative case, this is more or less a more convenient way to study real algebras.
Theorem: The functor sending a commutative complex -algebra to its subalgebra of self-adjoint elements is an equivalence of categories to the category of commutative -algebras.
Proof. The inverse of this functor sends an algebra to the tensor product with involution given by complex conjugation. That this is the inverse follows from the fact that for any element of a commutative complex -algebra we can write
which is precisely an element of tensor the self-adjoint subalgebra of ; moreover, morphisms out of are uniquely and freely determined by what they do to self-adjoint elements by the above.
Example. Let be the algebra of Laurent polynomials with involution given by . Identifying with the space of Fourier polynomials , we may identify the self-adjoint subalgebra with the space of trigonometric polynomials .
We may think of the above as being essentially about the Galois theory of the Galois extension ; more generally, if is any Galois extension, then there is a functor
and we may promote this to an equivalence of categories by attaching to the data of the action of the Galoi