Math 202B: Lecture 16

***Problems in this lecture due February 21***

In this lecture we will review the concrete algebras considered so far.

Function algebras. The archetypal commutative algebras: $\mathcal{F}(X)$ of functions on a finite set $X$ with all operations defined pointwise. We understand function algebras completely.

The elementary functions $\{E_x \colon x \in X\}$ are a basis of $\mathcal{F}(X)$ consisting of pairwise orthogonal selfadjoint idempotents: $E_x^*=E_x$ and $E_xE_y = \delta_{xy}E_x.$ In fact, this feature characterizes function algebras up to isomorphism.

Theorem: An algebra $\mathcal{A}$ is isomorphic to a function algebra if and only if it has a basis of pairwise orthogonal selfadjoint idempotents.

We have obtained a complete classification of subalgebras of function algebras.

Theorem: Subalgebras of $\mathcal{F}(X)$ are in bijection with partitions of $X$ . The correspondence is given explicitly by

$\mathcal{A}(\mathfrak{p})=\{A \in \mathcal{F}(X) \colon A \text{ constant on the blocks of }\mathfrak{p}\}.$

Corollary: Every subalgebra of a function algebra is isomorphic to a function algebra.

We have a complete understanding of ideals in function algebras.

Theorem: Ideals in $\mathcal{F}(X)$ are in bijection with subsets of $\mathcal{X}.$ The correspondence is given explicitly by

$\mathcal{J}(S) = \{A \in \mathcal{F}(X) \colon A \text{ vanishes on }S\}.$

We have a complete understanding of states on function algebras.

Theorem: States on $\mathcal{F}(X)$ are in bijection with probability functions on $X.$ The correspondence is given explicitly by

$\tau_P(A) = \sum\limits_{x \in X} A(x)P(x).$

Corollary: Faithful states on $\mathcal{F}(X)$ are in bijection with nonvanishing probability functions on $X.$

Corollary: Homomorphisms $\mathcal{F}(X) \to \mathbb{C}$ are in bijection with points of $X.$ The correspondence is given explicitly by

$\varphi_x(A)=A(x), \quad A \in \mathcal{F}(X).$

Endomorphism algebras. The archetypal noncommutative algebras: $\mathrm{End}(V)$ is the algebra of linear operators on a finite-dimensional Hilbert space $V$ .

If $X \subset V$ is an orthonormal basis, a vector space basis of $\mathrm{End}(V)$ is given by the elementary operators

$E_{yx}v=y\langle x,v\rangle, \quad x,y \in X,\ v \in V.$

The relations governing the elementary operators are

$E_{yx}^*=E_{xy} \quad\text{and}\quad E_{zy}E_{xw} = \langle y,x\rangle E_{zw}.$

In particular, $\{E_{xx} \colon x \in X\}$ spans a subalgebra $\mathcal{D}(X)$ isomorphic to $\mathcal{F}(X).$

Theorem: Every MASA in $\mathrm{End}(V)$ is of the form $\mathcal{D}(X)$ for some orthonormal basis $X \subset V.$

Corollary: All abelian subalgebras of $\mathrm{End}(V)$ have dimension at most $\dim V.$

Our understand of abelian subalgebras of $\mathrm{End}(V)$ is the following.

Theorem: Every abelian subalgebra of $\mathrm{End}(V)$ is isomorphic to a subalgebra of $\mathcal{F}(X)$ for some orthonormal basis $X \subset V.$ In particular, all abelian subalgebras of $\mathrm{End}(V)$ are isomorphic to function algebras.

At present, this is all we know about subalgebras of $\mathrm{End}(V)$ . A major reason for wanting to know more is the following.

Theorem: Every von Neumann algebra $(\mathcal{A},\tau)$ is isomorphic to a subalgebra of $\mathrm{End}(V)$ for some Hilbert space $V.$

On the bright side, our understanding of states and traces on $\mathrm{End}(V)$ . Let $X \subset V$ be an orthonormal basis and define a corresponding linear functional on $\mathrm{End}(V)$ by

$\mathrm{Tr}(A) = \sum\limits_{x \in X} \langle x,Ax \rangle.$

Theorem: All traces on $\mathrm{End}(V)$ are scalar multiples of $\mathrm{Tr}.$

Corollary: If $\dim V>1,$ there are no algebra homomorphisms $\mathrm{End}(V) \to \mathbb{C}.$

An operator $P \in \mathrm{End}(V)$ is called nonnegative if $P=Q^*Q$ for some $Q \in \mathrm{End}(V)$ . A nonnegative operator such that $\mathrm{Tr}(P)=1$ is called a density operator.

Theorem: States on $\mathrm{End}(V)$ are in bijection with density operators. The correspondence is given explicitly by $\tau_P(A)=\mathrm{Tr}(AP).$

Convolution algebras. The convolution algebra $\mathcal{C}(G)$ of a finite group $G$ coincides with the function algebra $\mathcal{F}(G)$ as a vector space, but conjugation and multiplication are instead defined by

$E_g^*=E_{g^{-1}} \quad\text{and}\quad E_gE_h=E_{gh}, \quad g,h \in G$ .

In particular, $\{E_g \colon g \in G\}$ is a basis of $\mathcal{C}(G)$ consisting of unitary elements rather than orthogonal idempotents.

Our understanding of convolution algebras is quite basic at present. For example, we know how to associate a subalgebra $\mathcal{C}(H)$ of $\mathcal{C}(G)$ to a subgroup $H \leq G,$ but we do not know much else about subalgebras of $\mathcal{C}(G).$

We did however manage to classify states and traces on $\mathcal{C}(G).$ A function $P \in \mathcal{C}(G)$ is called nonnegative if for any finite choice $\alpha_1,\dots,\alpha_n \in \mathbb{C}$ and $g_1,\dots,g_n \in G$ we have

$\sum\limits_{i,j=1}^n \overline{\alpha_i}\alpha_jP(g_i^{-1}g_j) \geq 0.$

Theorem: States on $\mathcal{C}(G)$ are in bijection with nonnegative functions in $\mathcal{C}(G)$ . The correspondence is given explicitly by

$\tau_P(A) =\sum\limits_{g \in G}A(g)P(g),$

and $\tau_P$ is a trace if and only if $P$ is a central function.

The only function $P \colon G \to \mathcal{C}$ which is both a probability function and a nonnegative function in the above sense is $E_e,$ and it is also a class function. Furthermore, the corresponding tracial state

$\tau_{E_e}(A) =A(e)$

is faithful. Therefore, $(\mathcal{C}(G),\tau_{E_e})$ is a von Neumann algebra and we can conclude the following.

Theorem: The convolution algebras $\mathcal{C}(G)$ is isomorphic to a subalgebra of $\mathrm{End}(V)$ for some Hilbert space $V.$ Furthermore, if $G$ is abelian then $\mathcal{C}(G)$ is isomorphic to a function algebra.

We also have a classification of homomorphisms from $\mathcal{C}(G)$ to $\mathbb{C}.$

Theorem: A state $\tau_P$ on $\mathcal{C}(G)$ is an algebra homomorphism if and only if $P \in \mathcal{C}(G)$ is a group homomorphism from $G$ to the unit circle $\mathbb{U} \subset \mathbb{C}.$

Problem 16.1. Show that there exists at least one algebra homomorphism $\mathcal{C}(G) \to \mathbb{C}.$ Given an example of a group for which this minimum is achieved.

Class algebras. The center of $\mathcal{C}(G)$ is denoted $\mathcal{K}(G)$ and it consists precisely of functions constant on conjugacy classes of $G$ . Therefore, we call $\mathcal{K}(G)$ the class algebra of $G.$ Writing $\{C_\alpha \colon \alpha \in \Lambda\}$ for the set of conjugacy classes in $G$ , the functions

$K_\alpha =\sum\limits_{g \in C_\alpha} E_g, \quad \alpha \in \Lambda,$

form a basis of $\mathcal{K}(G).$ This basis is orthogonal in the $L^2$ -scalar product,

$\langle K_\alpha,K_\beta \rangle = \delta_{\alpha\beta}|K_\alpha|.$

Problem 16.2. Show that $\tau_{E_e}(K_\alpha K_\beta)$ is equal to the number of solutions to the equation $xy=e$ in $G$ such that $x \in C_\alpha$ and $y \in C_\beta.$