f 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 define the absolute value (not to be confused with the norm ) by taking the pointwise absolute value in .
only needs to be defined on the spectrum of in order for to be well-defined. Thus if is self-adjoint then we can make sense of for any continuous . If is self-adjoint and has positive spectrum then we can make sense of, for example, and .
As a more specific application (although I do not think I need the full power of the continuous functional calculus to conclude this), recall that a one-parameter subgroup of a topological group is a continuous group homomorphism .
Theorem: The map assigning an element of a C*-algebra the function is a bijection from self-adjoint elements of to norm-continuous one-parameter subgroups of the unitary group of .
Proof. If is self-adjoint, lands in the commutative C*-subalgebra generated by , and any one-parameter subgroup of takes values in a commutative C*-subalgebra, so we may assume without loss of generality that is commutative. If is self-adjoint, then , thought of as a one-parameter family of elements of , is a one-parameter subgroup by the boundedness of , and taking adjoints shows that it is unitary. Conversely, if is a one-parameter subgroup of , then for sufficiently small , as a function on , will only take values in a ball of small radius around by norm-continuity. On a sufficiently small such ball we can define a branch of the logarithm, and then is a one-parameter family of functions such that . Since is continuous in , it follows that for some , and by unitarity is self-adjoint.
The above theorem describes the exponential map from the unitary Lie algebra
of skew-adjoint elements of (which are precisely the elements where is self-adjoint) to the unitary group . However, it is stated for self-adjoint elements so that it becomes a version of Noether’s theorem for C*-algebras, with self-adjoint elements of a C*-algebra taking the role of observables of some quantum system (the noncommutative space associated to ) and one-parameter subgroups of taking the role of one-para