 . Then has real spectrum.
This sum converges absolutely for all and we have . Standard analytic arguments show that
and we know that by the above. It follows that if is any morphism of Banach algebras, it is in particular a weak contraction, so . But and ; combining these two conditions we conclude that has zero imaginary part.
We now have enough to prove the following remarkable result.
Theorem (commutative Gelfand-Naimark): Let be a commutative C*-algebra. Then the Gelfand representation is an isomorphism of C*-algebras.
In particular, commutative C*-algebras are semiprimitive.
Proof. It remains to show that 1) the Gelfand representation respects , 2) the Gelfand representation is an isometry (hence ) and in particular injective, 3) the Gelfand representation is surjective.
1): Let where are self-adjoint (recall that every element of is uniquely of this form). If is a morphism, then are real by the lemma above, hence
2): This follows from and 1), since the Gelfand representation preserves spectral radius. If it follows now that , hence , hence the Gelfand representation is injective.
3): By 2), may be identified with a closed subalgebra of with norm the induced norm which is closed under complex conjugation, and by definition of the maximal spectrum it separates points. By the complex Stone-Weierstrass theorem, it then follows that .
Remember that when we exhibited a contravariant equivalence between compact Hausdorff spaces and certain -algebras we did not specify what algebras arose in this way. The Gelfand-Naimark theorem supplies the answer: they are precisely the self-adjoint subalgebras of commutative C*-algebras. Combined with results we already know about this equivalence, we conclude the following.
Gelfand duality: The functors form a contravariant equivalence between commutative C*-algebras and compact Hausdorff spaces.
Corollary: Let be a topological space. Then is the Stone–Cech compactification of .
Gelfand duality is a very suggestive theorem: it suggests that doing topology with compact Hausdorff spaces is equivalent to doing algebra with commutative C*-algebras. The idea that noncommutative C*-algebras therefore constitute a generalization of compact Hausdorff spaces is the basis of the noncommutative topology program.
But even without getting into such issues, Gelfand duality also has relatively concrete applications. For example, it implies a weak form of the spectral theorem: if is a normal operator on a Hilbert space , then considering the commutative C*-algebra generated by it follows that is isomorphic, in an appropriate sense, to a multiplication operator
where is a continuous complex-valued function on a compact Hausdorff space (namely the image of in ). Ideally we would want to be an element of and we would want the identification between and to be unitary, and it seems that we need to find an appropriate Borel measure on to say this. However, even without such a measure we still see that is diagonalizable in a suitable sense.
If is finite-dimensional, then so is , and this is possible if and only if is finite with the discrete topology; call its points . Note that must be the number of distinct eigenvalues of and the points may be identified with the set (not multiset) of eigenvalues of . Moreover, contains distinguished idempotents , the indicator functions of the points . These elements are self-adjoint idempotents in , so they must in fact be projections, and they are precisely the projections onto the eigenspaces of . So the spectral theorem provided by Gelfand duality really does recover the ordinary finite-dimensional spectral theorem.
Another application is as follows. Let be a normal element of a C*-algebra . The closure of defines a commutative C*-subalgebra of (the minimal such thing). By Gelfand duality, . Now, the latter space is closed under composition by any continuous function , Hence if is any such function, we can define a continuous functional calculus which allows us to apply continuous functions to normal elements of and to get other elements of . For example, we can defi