C*-Algebras: Spectral Theory and the Gelfand Representation

0. Key points

• The spectrum $\sigma(a)$ of an element $a$ of a complex Banach algebra is always a nonempty compact set. All the analysis of the subject is concentrated in this single point, which is proved through Liouville’s theorem; everything that follows is a consequence of it.
• The axiom that defines a $C^*$-algebra is the single identity $\|a^*a\| = \|a\|^2$. From it alone the norm is determined uniquely by the $*$-algebraic structure. This is the starting point of noncommutative geometry: all the geometric information already sits inside the algebra.
• A commutative $C^*$-algebra is nothing other than $C_0(X)$ for a locally compact Hausdorff space $X$, and $X$ can be recovered from the algebra as its character space (the commutative Gelfand–Naimark theorem). “Locally compact Hausdorff space” and “commutative $C^*$-algebra” carry exactly the same information.
• Noncommutative geometry keeps the algebraic side of this dictionary, discards commutativity, and reads a general $C^*$-algebra as a noncommutative topological space.
• The continuous functional calculus $C(\sigma(a)) \to A$ for a normal element $a$ is the common foundation for all the later tools: positivity, projections, unitaries, and hence $K$-theory and spectral triples.

1. Motivation: why make an algebra of functions noncommutative?

When we study a topological space $X$, we very often study not $X$ itself but the ring $C(X)$ of continuous functions on it. In algebraic geometry this policy is carried through completely: an affine variety carries exactly the same information as its coordinate ring. Can the same be said of topological spaces? That is, can $X$ be recovered from $C(X)$?

In the late 1930s Gelfand built a framework that answers this question. His 1941 paper “Normierte Ringe” introduced the construction that extracts a character space from a commutative Banach algebra. Then in 1943 Gelfand and Naimark showed that imposing the single identity $\|a^*a\| = \|a\|^2$ on a Banach algebra with involution already forces it, in the commutative case, to be $C_0(X)$ and nothing else. The space $X$ is recovered from the algebra completely. A space no longer needs to be presented as a set of points: the algebra of functions is enough.

At about the same time, quantum mechanics was demanding algebras from a different direction. Observables do not commute — position and momentum satisfy $[q, p] = i\hbar$ — so the classical picture of a “point of phase space” simply fails. This is why von Neumann began the study of operator algebras. And here a natural question arises. If a commutative $C^*$-algebra is a space, what is a noncommutative one?

Connes’ answer is that it too is a space, only a noncommutative one. This point of view acquires its force exactly where the classical construction collapses. The leaf space of a foliation and the orbit space of a group action, given the quotient topology, fail to separate points, and almost no continuous functions survive on them. The corresponding noncommutative $C^*$-algebras, by contrast, are rich; $K$-theory and index theorems work on them. For details see Basics of C*-algebras, and in particular the orbit space of the circle under an irrational rotation（Example 7.1）.

This article assembles the foundations. It also answers the question of why one works with $C^*$-algebras rather than Banach algebras. The answer is Corollary 4.3. The norm of a Banach algebra is extra data supplied from outside the algebra, whereas a $C^*$-norm is determined uniquely by the algebraic structure alone. It is precisely because the topology comes along automatically that one may define a noncommutative space by an algebra.

The logical flow of this article. We start from Banach algebras at the top left and descend in a single line to the continuous functional calculus at the bottom right.

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