Math 202B: Lecture 14

We have been comparing and contrasting the function algebra $\mathcal{F}(G)$ of a finite group $G$ with its convolution algebra $\mathcal{C}(G).$ These objects are the same as vector spaces, but very different as algebras. In particular, we understand what is happening inside $\mathcal{F}(G)$ very well, in the sense that we know all of its subalgebras, whereas our understanding of subalgebras of $\mathcal{C}(G)$ is limited to those coming from subgroups $H \leq G$ together with the class algebra $\mathcal{K}(G) =Z(\mathcal{C}(G)).$

Today we will compare and contrast $\mathcal{F}(G)$ and $\mathcal{C}(G)$ through the lens of states and traces. The starting point is the same in both cases: a very simple lemma identifying linear functionals on a function algebra $\mathcal{F}(X)$ with functions in $\mathcal{F}(X).$

Lemma 14.1. For any set $X$ , linear functionals on $\mathcal{F}(X)$ are in linear bijection with functions in $\mathcal{F}(X)$ .

Proof: For any functional $\varphi \colon \mathcal{F}(X) \to \mathbb{C}$ , linear or not, we get a corresponding function $F_\varphi \in \mathcal{F}(X)$ defined by

$F_\varphi(x) = \varphi(E_x), \quad x \in X.$

If $\varphi$ is a linear functional on $\mathcal{F}(X)$ , then it is uniquely determined by its values on the basis $\{E_x \colon x \in X\}$ of elementary functions, hence $\varphi$ is uniquely determined by $F_\varphi.$ Conversely, if $F \in \mathcal{F}(X)$ is any function, we can define a linear functional on $\mathcal{F}(X)$ by

$\varphi_F(E_x) = F(x), \quad x \in X.$

It is clear that the maps $\varphi \mapsto F_\varphi$ and $F \mapsto \varphi_F$ are inverses of one another. $\square$

Recall our classification of states on $\mathcal{F}(X)$ , obtained using Lemma 14.1.

Theorem 14.2. The following are equivalent:

$P$ is a probability function on $X.$
$\tau_P$ is a state on $\mathcal{F}(X)$ .

In fact, we know a few enhancements of Theorem 14.2: first, $\tau_P$ is a faithful state if and only if $P(x)>0$ for all $x \in X$; second, $\tau_P$ is an algebra homomorphism if and only if $P=E_x$ for some $x \in X.$ In particular, all homomorphisms $\mathcal{F}(X) \to \mathbb{C}$ are evaluation at a point, i.e. maps of the form $A \mapsto A(x_0)$ for a particular $x_0 \in X.$

As vector spaces, $\mathcal{F}(G)$ and $\mathcal{C}(G)$ are not just isomorphic they are equal, hence Lemma 14.1 applies verbatim. However, since $\mathcal{F}(G)$ and $\mathcal{C}(G)$ are quite different as algebras, the classification of states on $\mathcal{C}(G)$ is going to be quite different. To prepare yourself for this, solve the following problem.

Problem 14.1. Let $P \in \mathcal{C}(G)$ be a probability function. Prove that the corresponding linear functional $\varphi_P$ is a state on $\mathcal{C}(G)$ if and only if $P=E_e$ is the indicator function of the group identity $e \in G.$ Conclude that in this case $\varphi_P(A) = A(e)$ is evaluation at $e,$ but that this is not an algebra homomorphism $\mathcal{C}(G) \to \mathbb{C}.$

Now let us determine which functions $P \in \mathcal{C}(G)$ correspond to states on $\mathcal{C}(G)$ . The normalization condition is straightforward: since the multiplicative identity $I \in \mathcal{C}(G)$ is $I=E_e,$ we have

$\tau_P(I)=1 \iff \tau_P(E_e)=1 \iff P(e)=1.$

Now let us consider what property $P \in \mathcal{C}(G)$ must have in order for $\tau_P$ to be a nonnegative functional. For an arbitrary function

$A=\sum\limits_{g \in G} \alpha_g E_g,$

we have

$A^*=\sum\limits_{g \in G} \overline{\alpha_g}E_{g^{-1}},$

so that

$\tau_P(A^*A) = \sum\limits_{g,h \in G} \overline{\alpha}_g\alpha_hP(g^{-1}h).$

Hence if we pick an ordering $g_1,\dots,g_n$ of $G$ , and write $\alpha_i=\alpha_{g_i},$ this becomes

$\tau_P(A^*A) = \sum\limits_{i,j=1}^N \overline{\alpha_i}\alpha_j P(g_i^{-1}g_j).$

Definition 14.3. A complex-valued function $P$ on a group $G$ is said to be nonnegative if for any $n \in \mathbb{N},$ any $\alpha_1,\dots,\alpha_n \in \mathbb{C},$ and any $g_1,\dots,g_n \in G$ we have

$\sum\limits_{i,j=1}^n \overline{\alpha}_i\alpha_j P(g_i^{-1}g_j) \geq 0.$

Problem 14.2. Prove $P$ is a nonnegative function on $G$ if and only if, for all $n \in \mathbb{N}$ and $g_1,\dots,g_n \in G$ the matrix $[P(g_i^{-1}g_j)]_{i,j=1}^n$ is Hermitian and has nonnegative eigenvalues. What is this matrix if $P=E_e$ ?

With the above in place, we can state our classification of states on the convolution algebra $\mathcal{C}(G)$ of a finite group $G$ as follows.

Theorem 14.4. The following are equivalent:

$P$ is a nonnegative function on $G$ ;
$\tau_P$ is a state on $\mathcal{C}(G).$

The classification of states on $\mathcal{C}(G)$ is even more straightforward.

Theorem 14.5. The following are equivalent:

$P$ is a central function on $G$ ;
$\tau_P$ is a trace on $\mathcal{C}(G)$ .

Proof: Suppose $P \in \mathcal{C}(G)$ is central. We know from last lecture that this is equivalent to $P(gh)=P(hg).$ Thus,

$\tau_P(E_gE_h)=P(gh)=P(hg)=\tau_P(E_hE_g).$

Conversely, if $\tau_P$ is a trace on $\mathcal{C}(G)$ then

$P(gh)=\tau_P(E_{gh})=\tau_P(E_gE_h)=\tau_P(E_hE_g) = \tau_P(E_{hg})=P(hg),$

whence $P$ is central. $\square$

We now have a complete classification of states and traces on the convolution algebra $\mathcal{C}(G)$ of a finite group $G.$ Namely, every linear functional on $\mathcal{C}(G)$ has the form

$\tau_P(E_g)=P(g), \quad g \in G$

for some function $P \in \mathcal{C}(G)$ . We know that $\tau_P$ is a state if and only if $P(e)=1$ and $[P(g^{-1}h)]_{g,h \in G}$ is a nonnegative Hermitian matrix. We know that $P$ is a trace if and only if $P(gh)=P(hg)$ is a central function on $G.$

Math 202B: Lecture 14

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