Math 202: Function algebras

The function algebra of a finite set X is the set \mathcal{F}(X) of functions A \colon X \to \mathbb{C} with operations defined pointwise. A basic family of functions are the elementary functions,

E_x(y)=\delta_{xy}, \quad x,y \in X.

Theorem F1. The elementary functions are nonzero pairwise orthogonal projections.

Combining Theorem F1 with Theorem A3, we get that the set \{E_x \colon x \in X\} is a basis of \mathcal{F}(X).

The algebra \mathcal{F}(X) can be understood completely. In particular, subalgebras can be explicitly classified. A partition of X is a set \mathfrak{p}=\{P_1,\dots,P_m\} of nonempty disjoint subsets of X whose union is X. The elements P_i of \mathfrak{p} are called its blocks. The set \mathfrak{P}(X) of partitions of X is partially ordered set under the refinement order, where \mathfrak{p} \leq \mathfrak{q} if and only if \mathfrak{q} can be obtained from \mathfrak{p} by breaking blocks. Associated to every \mathfrak{p} \in \mathfrak{P}(X) is the subalgebra \mathcal{A}(\mathfrak{p}) consisting of functions constant on the blocks of \mathfrak{p}. Note that \mathcal{A}(\mathfrak{p}) is isomorphic to \mathcal{F}(\mathfrak{p}).

Theorem F2. The set \{\mathcal{A}(\mathfrak{p}) \colon \mathfrak{p} \in \mathfrak{P}(X)\} is the complete collection of subalgebras of \mathcal{F}(X). Moreover, \mathcal{A}(\mathfrak{p}) \leq \mathcal{A}(\mathfrak{q}) if and only if \mathfrak{p}\leq \mathfrak{q}.

This classification theorem can be proved in several ways. A nice argument uses the finite-dimensional Stone-Weierstrass theorem. A subalgebra \mathcal{A} \leq \mathcal{F}(X) is said to separate points if, for every \{x,y\} \subseteq X, there exists a function A \in \mathcal{A} such that A(x) \neq A(y).

Theorem F3. If \mathcal{A} \leq \mathcal{F}(X) separates points, then \mathcal{A}=\mathcal{F}(X).

There is a parallel classification of ideals in \mathcal{F}(X). Let us order the set \{S \subseteq X\} of all subsets of X by inclusion. Associated to each S \subseteq X is the ideal

\mathcal{I}(S) = \{A \in \mathcal{F}(X) \colon A(x)=0 \text{ for all }x \in S\}.

Theorem F4. The set \{\mathcal{I}(S) \colon S \subseteq X\} is the complete collection of ideals in \mathcal{F}(X). Moreover, \mathcal{I}(S) \subseteq \mathcal{I}(T) if and only if T \subseteq S.

Just like Theorem F2 shows that we cannot discover any new algebras by looking at subalgebras of functional algebras, Theorem F4 reveals that we cannot discover any new algebras by taking quotients of function algebras.

Theorem F5. For any S \subseteq S, the quotient algebra \mathfrak{F}(X)/\mathcal{I}(S) is isomorphic to the function algebra \mathcal{F}(S).

Now let us classify von Neumann algebra structures on \mathcal{F}(X). This means that we will find all Frobenius scalar products on \mathcal{F}(X); since \mathcal{F}(X) is commutative there is no distinction between left-Frobenius and right-Frobenius. Equivalently, we classified faithful states on \mathcal{F}(X), all of which are tracial. A function P \in \mathcal{F}(X) is called a probability distribution if P(x) \geq 0 for all x \in X, and \sum_{x \in X} P(x)=1.

Theorem F6. States \sigma on \mathcal{F}(X) are in bijection with probability distributions on X: every state is of the form \sigma(A) = \sum_{x \in X} A(x)P(x) for a unique probability distribution P. Moreover, \sigma is faithful if and only if P is nonvanishing.

Theorem F6 is equivalent to the statement that all Frobenius scalar products are of the form

\langle A,B \rangle = \sum\limits_{x \in X} \overline{A(x)}B(x)P(x)

for a unique probability distribution P having full support in X.

Going beyond understanding the inner workings of a given function algebra \mathcal{F}(X), we can classify all algebras \mathcal{A} which are isomorphic to a function algebra.$ Clearly, any such \mathcal{A} must be commutative. A basis of \mathcal{A} consisting of pairwise orthogonal projections is called a Fourier basis.

Theorem F7. An algebra \mathcal{A} is isomorphic to a function algebra if and only if it admits a Fourier basis.

Theorem F7 is very important, and so is its proof. The fact that \mathcal{F}(X) admits a Fourier basis is clear (the elementary functions). Conversely, suppose that \mathcal{A} is an abstract algebra in which we are able to find a Fourier basis \{F^\lambda\colon \lambda \in \Lambda\} indexed by some finite set \Lambda. Then, every element A \in \mathcal{A} can be expanded in this basis,

A = \sum\limits_{\lambda \in \Lambda}\widehat{A}(\lambda)F^\lambda

and the coefficients in this expansion define a function \widehat{A} \in \mathcal{F}(\Lambda). The mapping

\Phi \colon \mathcal{A} \longrightarrow \mathcal{F}(\Lambda)

defined by \Phi(A) = \widehat{A} is an algebra isomorphism called the Fourier transform. The “the” here is justified by the following.

Theorem F8. The only Fourier basis is \mathcal{F}(X) is the basis \{E_x \colon x \in X\} of elementary functions.

Theorem F8 means that if a Fourier transform \Phi \colon \mathcal{A} \to \mathcal{F}(\Lambda) exists, then it is unique up to relabeling the label set \Lambda.

Published by Jonathan Novak

Mathematician at UC San Diego.

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