# C*-Algebras: Spectral Theory and the Gelfand Representation

> Spectra in Banach algebras are nonempty and compact; from this we derive the spectral radius formula, identify commutative C*-algebras with C0(X), and build the functional calculus.
> https://rikai.mugen-giken.com/en/mathematics/operator-algebras/c-star-algebras

## 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](/en/mathematics/operator-algebras/c-star-algebras), and in particular <Ref to="mathematics/operator-algebras/c-star-algebras#ex-irrational-rotation" text="the orbit space of the circle under an irrational rotation" />.

This article assembles the foundations. It also answers the question of why one works with $C^*$-algebras rather than Banach algebras. The answer is <Ref to="cor-norm-unique" />. 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.

<Figure caption="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.">
<Mermaid code={`flowchart TD
  BA["Banach algebra"] --> SPEC["the spectrum is a nonempty compact set"]
  SPEC --> RAD["spectral radius formula"]
  BA --> INV["introduce an involution"]
  INV --> CSTAR["C*-identity"]
  RAD --> NORMAL["for normal elements, norm = spectral radius"]
  CSTAR --> NORMAL
  NORMAL --> UNIQ["uniqueness of the C*-norm"]
  BA --> GEL["Gelfand transform"]
  GEL --> GN["commutative Gelfand–Naimark theorem"]
  NORMAL --> GN
  GN --> FC["continuous functional calculus"]
  FC --> NCG["towards K-theory and spectral triples"]`} />
</Figure>

(The rest of this chapter is available to purchasers.)