Banach algebras, the Gelfand representation, and the commutative Gelfand-Naimark theorem

Banach algebras abstract the properties of closed algebras of operators on Banach spaces. Many basic properties of such operators have elegant proofs in the framework of Banach algebras, and Banach algebras also naturally appear in areas of mathematics like harmonic analysis, where one writes down Banach algebras generalizing the group algebra to study topological groups.

Today we will develop some of the basic theory of Banach algebras, our goal being to discuss the Gelfand representation of a commutative Banach algebra and the fact that, for commutative C*-algebras, this representation is an isometric isomorphism. This implies in particular a spectral theorem for self-adjoint operators on a Hilbert space.

This material can be found in many sources; I am working from Dales, Aiena, Eschmeier, Laursen and Willis’ Introduction to Banach Algebras, Operators, and Harmonic Analysis.

Below all vector spaces are over , all algebras are unital, and all algebra homomorphisms preserve units unless otherwise stated. In the context of Banach algebras, the last two assumptions are not standard, but in practice non-unital Banach algebras are studied by adjoining units first, so we do not lose much generality.

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Hilbert spaces (and dagger categories)

Hilbert spaces are a particularly nice class of Banach spaces. They axiomatize ideas from Euclidean geometry such as orthogonality, projection, and the Pythagorean theorem, but the ideas apply to many infinite-dimensional spaces of functions of interest to various branches of mathematics. Hilbert spaces are also fundamental to quantum mechanics, as vectors in Hilbert spaces (up to phase) describe (pure) states of quantum systems.

Today we’ll develop and discuss some of the basic theory of Hilbert spaces. As with the theory of Banach spaces, there are (at least) two types of morphisms we might want to talk about (unitary operators and bounded operators), and we will discuss an elegant formalism that allows us to talk about both. Things written by John Baez will be cited excessively.

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Banach spaces (and Lawvere metrics, and closed categories)

One annoying feature of the abstract theory of vector spaces, and one that often trips up beginners, is that it is not possible to make sense of an infinite sum of vectors in general. If we want to make sense of infinite sums, we should probably define them as limits of finite sums, so rather than work with bare vector spaces we need to work with topological vector spaces over a topological field, usually or (but sometimes fields like are also considered, e.g. in number theory). Common and important examples include spaces of continuous or differentiable functions.

Today we’ll discuss a class of topological vector spaces which is convenient to work with but which still covers many examples of interest, namely Banach spaces. The material in the first half of this post is completely standard and can be found in any text on functional analysis.

In the second half of the post we discuss a category of Banach spaces such that two Banach spaces are isomorphic in this category if and only if they are isometrically isomorphic but which still allows us to talk about bounded linear operators between Banach spaces, and to do this we briefly discuss Lawvere metrics; this material can be found on the nLab.

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ab, ba, and the spectrum

Let be two matrices. If don’t commute, then ; however, the two share several properties. If either or is invertible, then is conjugate to , so in particular they have the same characteristic polynomial.

What if neither nor are invertible? As it turns out, and still have the same characteristic polynomial, although they are not conjugate in general (e.g. we might have but nonzero). There are several ways of proving this result, which implies in particular that and have the same eigenvalues.

What if are linear transformations on an infinite-dimensional vector space? Do and still have the same eigenvalues in an appropriate sense? As it turns out, the answer is yes, and the key lemma in the proof is an interesting piece of “noncommutative high school algebra.”

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Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when two varieties are isomorphic, and so it’s worth describing carefully.

The geometric picture is perhaps clearest in the case where is a compact Hausdorff space and is its ring of functions. From this definition it follows that is a contravariant functor from the category of compact Hausdorff spaces to the category of -algebras (which we are assuming have identities). Explicitly, a continuous function

between compact Hausdorff spaces is sent to an -algebra homomorphism

in the obvious way: a continuous function is sent to a continuous function . The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that is an embedding because then one may identify with the restriction of to the image of . This restriction takes the form of a homomorphism whose kernel is the set of functions which are zero on , so it exhibits as a quotient of .

Question: Does every -algebra homomorphism come from a continuous function ?

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