# Introduction

In what follows, we always work over the field of complex numbers $\mathbb {C}$ . For purely algebraic considerations, the only key property of $\mathbb {C}$ that we use is that $\mathbb {C}$ is algebraically closed, i.e. that every non-constant polynomial with complex coefficients has at least one complex root. Furthermore, using $\mathbb {C}$ gives us the possibility to use complex topology, which is useful for illustrative purposes and also to make contact with complex or differential geometry.

The main point is the idea of "duality" between spaces and rings, or algebras, of functions on spaces. This duality is non-trivial in the sense that it exchanges difficult notions on one side with easy notions on the other side. It is also a contravariant operation: all the natural maps on one side go to natural maps on the other side going in the opposite direction.

The idea of duality between spaces and functions is a vague general idea that can be made precise in specific contexts.

Example

Let $V$ be a finite-dimensional vector space over $\mathbb {C}$ . Natural functions on $V$ are given by linear functions. The set of linear functions on $V$ is a finite-dimensional vector space $V^{*}$ called the "dual" of $V$ . It is possible to recover the space $V$ from its space of functions $V^{*}$ in the following way. Every $x\in V$ defines a linear function

${\begin{array}{ccccc}ev_{x}\colon &V^{*}&\longrightarrow &\mathbb {C} \\&f&\mapsto &f(x)\\\end{array}}$ called the "evaluation map at $x$ ". By dimension considerations, one shows that every linear function on $V^{*}$ is of this form, i.e. the natural linear map $V\rightarrow V^{**}$ given by the evaluation is an isomorphism.

Example

Gelfand-Naimark, classification of commutative $C^{*}$ -algebras.

Let $X$ be a Hausdorff compact topological space. We consider the space $C(X)$ of continuous complex-valued functions $f\colon X\rightarrow \mathbb {C}$ . It is a vector space over $\mathbb {C}$ , in general infinite dimensional. It also has a natural algebra structure (with unit) given by the product of functions, and a norm defined by $||f||=sup_{x\in X}|f(x)|$ . It is easy to show that for every $f$ and $g$ in $C(X)$ we have $||f.g||\leq ||f||.||g||$ , and that $C(X)$ with the norm $||.||$ is a complete normed vector space. In other words, $C(X)$ has a natural structure of a commutative "Banach algebra".

Furthermore, we define a map $f\mapsto f^{*}$ from $C(X)$ to itself by $f^{*}(x)={\overline {f(x)}}$ where $z\mapsto {\overline {z}}$ is the complex conjugation. The map $*$ is antilinear, it is a morphism of rings, and it satisfies obvious compatibilities with the norm, such that $||f.f^{*}||=||f||^{2}$ .

A Banach algebra with such an operation $*$ is called a "$C^{*}$ -algebra" and we have just shown how to associate to every Hausdorff compact topological space $X$ its $C^{*}$ -algebra of continuous functions $C(X)$ .

Every $x\in X$ defines a $*$ -homomorphism, i.e. a continuous algebra homomorphism compatible with the operation $*$ ,

${\begin{array}{ccccc}ev_{x}\colon &C(X)&\longrightarrow &\mathbb {C} \\&f&\mapsto &f(x)\\\end{array}}$ called the "evaluation map at $x$ ". It is possible to show that any $*$ -homomorphism is of this form and so that a compact topological space $X$ can be recovered from the $C^{*}$ -algebra $C(X)$ .

One can also show that any commutative $C^{*}$ -algebra with unit is of the form $C(X)$ for some Hausdorff compact topological space $X$ called the "spectrum" of the $C^{*}$ -algebra.

This means that the map $X\mapsto C(X)$ identifies a notion of "space", the notion of Hausdorff compact topological space, with a notion of "algebra of functions", the notion of commutative $C^{*}$ -algebra with unit.

Exercise

Show that if one forgets the operation $*$ on $C(X)$ , one gets something like a complexification of $X$ . For example, let $X=S^{1}$ be a circle parametrized by some angle $\theta \in \mathbb {R} /2\pi \mathbb {Z}$ and let $R$ be the algebra over $\mathbb {C}$ of finite Fourier series $f=\sum _{n\in \mathbb {Z} }a_{n}e^{in\theta }$ on $S^{1}$ . Show that the space of $*$ -homomorphisms $R\rightarrow \mathbb {C}$ can be identified with $S^{1}$ whereas the space of algebra homomorphisms $R\rightarrow \mathbb {C}$ can be identified with $\mathbb {C} ^{*}$ .

Hint: Write $f=\sum _{n\in \mathbb {Z} }a_{n}z^{n}$ so that $R=\mathbb {C} [z,z^{-1}]$ .