ver 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 Galois group on the left factor; taking the -invariant subalgebra then gives the inverse functor. None of this requires that the -algebras or -algebras in question are commutative.
However, noncommutative complex -algebras do not fit into this formalism since the involution reverses the order of multiplication. In particular, the self-adjoint elements no longer form a subalgebra; if are self-adjoint but don’t commute then
The self-adjoint elements are instead closed under the anticommutator
and axiomatizing this observation leads to the theory of Jordan algebras. However, we won’t be concerned much with anticommutators; if are observables in a quantum system, we generally prefer to look at the observable , for example when is position and is momentum, in view of Noether’s theorem.
We are now ready for the following definition axiomatizing the properties of closed -closed subalgebras of . A C*-algebra is a Banach algebra equipped with an involution making it a complex -algebra and also satisfying the B*-identity
(See Wikipedia for the history here.) A morphism of C*-algebras is a morphism of Banach algebras which respects the involution.
(It is not obvious that the B*-identity is either useful or enough to capture everything we want. Happily, both are true.)
Proposition: Let be a Hilbert space. Then the algebra of bounded linear operators is a C*-algebra with involution the adjoint.
Proof. Let be such an operator. Then
and the conclusion follows.
Corollary: Any closed -closed subalgebra of is also a C*-algebra.
Example. Let be a bounded linear operator. Then generate a -closed subalgebra of whose closure is also -closed (this follows from the fact that taking adjoints preserves norms, hence is itself continuous with norm ), hence gives a C*-algebra. This C*-algebra is commutative if and only if is normal.
Example. Let be a collection of bounded linear operators on a Hilbert space which is -closed. Then their commutant is also -closed, hence is a C*-algebra. These C*-algebras are precisely the von Neumann algebras.
Exam