s 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.
Example. If is a compact Hausdorff space then is a commutative C*-algebra with the involution being complex conjugation. It may be regarded as a space of operators acting by multiplication on if is equipped with a suitably nice Borel measure, although I am not sure of the precise hypotheses we need for the operator norm to match the sup norm. At a minimum the measure of every open set must be positive.
We now need the following.
Black Box #2 (Beurling-Gelfand spectral radius formula): Let be an element of a Banach algebra . Then the spectral radius satisfies
This is a surprising result in light of the fact that the LHS depends only on the algebra structure of while the RHS is defined in terms of the norm on . The crux of the proof is the following version of the Cauchy integral formula, which we will not prove here:
for all . If , then it follows that
hence that for all . Since we already know that , the conclusion follows. Again, we defer discussion of the relevant complex analysis to a later post.
Corollary: Let be a self-adjoint element of a C*-algebra. Then .
Proof. The B*-identity gives , and iterating this gives
for all . Taking -th roots of both sides and taking limits, the conclusion follows by the spectral radius formula.
Corollary: Let be an element of a C*-algebra . Then . Consequently, .
Proof. The B*-identity and the above gives since is self-adjoint. Now, , and since is invertible if and only if is invertible, it follows that , hence , and the conclusion follows.
So the norm on a C*-algebra is already uniquely determined by its complex -structure. Moreover:
Corollary: Let be C*-algebras. Any complex -algebra morphism is a weak contraction; consequently, it is already a morphism of C*-algebras. Equivalently, the category of C*-algebras is a full subcategory of the category of complex -algebras.
Proof. If is invertible in then is invertible in . Consequently, , so . By the above, it then follows that .
Before we proceed, we will also need the following.
Lemma: Let be a self-adjoint element of a C*-algebra