The classical Wedderburn theorem, dating back to 1908, is a milestone in abstract algebra. This article is devoted to the proof the this beautiful theorem over arbitrary fields, after which we specialize to $\mathbb{C}$ and $\star$ -algebras.

The first part of this post will be a shameless copy of the lecture notes of a course in representation theory held in the spring of 2017 by Prof. Geir Ellingsrud at the university of Oslo and are readily available online. I restate the results here for convenience. The second part will be devoted to applications of these classical results to operator theory.

Table of Contents

Introduction

Some notation and a note on the endomorphism ring $End_A(V)$

In what follows, $A$ is a finite dimensional algebra over any field $k$ and for an $A$ -module $V$ we set

$D = End_A(V).$

Unless $A$ is commutative, the multiplication by $a$ maps

$M_a: V\to V \qquad v\mapsto av$

are not $A$ -linear morphisms of $V$ , but they do commute (by definition) with every element in the ring $D$ . The natural map

(1) $\begin{equation*}A\to D = End_A(V) \qquad a\mapsto M_a\end{equation*}$

is the ring homomorphism that determines the module structure of $V$ over $A$ . We can however treat $V$ as a $D$ -module in stead in the obvious way by setting $\phi \cdot v := \phi (v)$ for all $\phi \in D$ and $v\in V$

Note that (by Schur’s lemma) if $V$ is irreducible, $D$ is a division ring as every element is either invertible of the zero morphism. Modules over division rings are particularly well behaved and can all be treated “essentially” like vector spaces in the sense that they are all free (i.e. have a basis), meaning that after a choice of basis we may write

$End_A(V) \simeq M_{n\times n} (D^{op}).$

The ‘op’ supscript stands for the oposite algebra/ring and is just the algebra with multiplication reversed. If in doubt, work out the details to check that we have to do this. We may get away with substituting $D$ by the scalar field $k$ in some cases, like the following (not the similarities with the Gelfand-Mazur theorem) –

If $k$ is algebraically closed field, $A$ is a finiti dimensional $k$ -algebra and $V$ is an irreducible $A$ -module, then
$D = k$

Lastly, note that the ring $End_{D_W}(W)$ is an $A$ -module in its own right with pointwise defined operations (where $D_W = End_A(W)$ ). If $A$ is an algebra of finite dimension over $k$ , $W$ an irreducible $A$ -module and $w_1, ..., w_n \in W$ is a basis for $W$ over $D_W$ , then as $A$ -modules one has isomorphisms

(2) $\begin{equation*}End_{D_W}(W) = \underbrace{W\oplus...\oplus W}_n \qquad \phi \mapsto (\phi(w_i)).\end{equation*}$

Hence $End_{D_W}(W)$ is a semisimple $A$ -module. These modules are quite simple to work with, no pun intended. Let us look at some of their basic properties –

Sem-simple $A$ -modules

Given a semi-simple $A$ -module, $V = V_1 \oplus ... \oplus V_n$ we may group together all $V_i$ ‘s which are isomorphic to one another and write

$V = U_{V_{i_1}}(V) \oplus ... \oplus U_{V_{i_m}}(V)$

where $U_{W}(V) = W\oplus... \oplus W$ is the collection of all elements isomorphic to $W$ in the above decomposition of $V$ . The modules $U_{V_i}(V)$ are called the isotypic components of $V$ corresponding to the irreducible $A$ -module $V_i$ .

Since we don’t care about the exact decomposition of $V$ into irreducibles, it is customary to only work with isomorphism classes of irreducible $A$ -modules as the assignment $V\mapsto U_W(V)$ is functorial by (you guessed it..) Schur’s lemma. In summary, there is a “functorial” way to assign to a semisimple $A$ -module $W$ its isomorphism class using only the isomorphism classes of its decomposition. Namely

$V = \bigoplus_{W\in Irr(A)}U_W(V)$

where $V$ and $W$ are treated as isomorphism classes and the sum is finite since $V$ is semi-simple.

This is the content of the following theorem

Theorem (Isotypic decomposition theorem) Let $A$ be a ring and $V$ a semi-simple $A$ -module. Then $V$ has a canonical decomposition

$V = \bigoplus_{W\in Irr(A)} U_W(V)$

Submodules of semi-simple $A$ -modules are as well behaved as we may have hoped them to be. Any submodule $U\subset V= \bigoplus U_W(V)$ , with $U_W(V) = W_1\oplus... \oplus_{n_W}$ is a some direct sum of copies of $W_i$ , while morphisms of semi-simple respect the isotypic decomposition, meaning that for any semi-simple $A$ -modules $V$ and $V'$ we have

$Hom(V, V') = \prod Hom(U_W(V), U_W(V')).$

All these results follow more or less directly from Schur’s lemma.

Proof of Wedderburn’s theorem

Let’s first state the theorem in question –

Theorem 1 (Wedderburn’s theorem) Given a finite dimensional algebra $A$ over any field $k$ there is a natural surjective morphism

$A\to \prod_{W\in Irr(A)}End_{D_W}(W)$

Where the index runs over all isomorphism classes of irreducible $A$ -modules. It was later shown by Emil Artin that Wedderburn’s theorem holds also for infinite algebras $A$ provided it has finite length (as a module over itself). We will not prove this more general result here.

The proof of the theorem relies on several results known at the time, the first of which is –

Theorme (Jacobson’s density theorem) Let $V$ be a simple $A$ -module, and let $v_1, ..., v_n$ be vectors in $V$ which are linearly independent over $D = End_A(V)$ . For any vectors $w_1, ..., w_n$ in $V$ there exists an $a \in A$ with $w_i = av_i$

Proof

Define a map

$\Psi: A\to \underbrace{V\oplus ....\oplus V}_n\qquad a \mapsto (av_i).$

The image of this map determins an $A$ -submodule of $\underbrace{V\oplus ....\oplus V}_n$ and by Schur’s lemma must be of the form $\underbrace{V\oplus .... \oplus V}_m$ for some $m\leq n$ with complementary subbundle $U'$ such that

$\underbrace{V\oplus ....\oplus V}_n = U\oplus U'.$

The projection onto $U'$ is $D$ -linear and can thus be expanded as a matrix $(d_{ij})$ with entries in $D$ . Since $v_1, …, v_n$ are linearly independent, we have that for all $j$ , $\sum_id_{ij}(av_{j})=a\left(\sum_id_{ij}(v_{j})\right)= 0$ since this holds for every $a\in A$ and $V$ irreducible we must have that

$\sum_id_{ij}v_{j}=0 ~~ \forall j=1,...,n\Rightarrow d_{ij} =0 ~~\forall i,j=1,...,n\Rightarrow U' =0$

by linear independence. This proves the theorem.

Not the similarities with Kaddison transitivity theorem, which deals with irreducible representation $\pi: A\to B(H)$ of a C*-algebra $A$ , i.e. a topologically irreducible $A$ -module $H$ . In that setting $D$ -linear independence corresponds to the usual linear independence in $H$ since $D = End_A(H) = End_\mathbb{C}(H)$ – why? Big hint – $\pi$ is irreducible if and only if $\pi(A) ' = \mathbb{C}I$ (see Theorem 5.2.5 of [1]).

Anyway, the second result which we will use is the following –

Theorme (Burnside) Let $A$ be a finite dimensional algebra over a field $k$ and let $V$ be an irreducible module over $A$ . Then the map

$A \to End_D(V) \qquad a\mapsto M_a$

(see equation 1) is surjective.

Proof

The module $V$ is a finite module over $A$ by assumption, hence it is a
finite dimensional vector space over $k$ and a fortiori over the larger $D$ ( $k$ sits inside $D$ as scalars). Pick a basis $v_1,...,.v_n$ for $V$ over $D$ . For every $D$ -linear map $\phi: V \to V$ applying the density theorem to the vectors $\phi(v_1), ..., \phi(v_n)$ one sees that there exists an element $a$ from $A$ so that $av_i = \phi(v_i)$ for $1 \leq i \leq n$ . Thus the two $D$ -linear maps $\phi$ and multiplication-by- $a$ agree on a basis and therefore $\phi(v) = av$ for all $v\in V$ .

In particular if the algebra $A$ is known to be simple we have that

$A\simeq End_D(V).$

If we collect these maps into a map $A\mapsto \prod_{W\in Irr(A)}End_{D_W}(W)$ hoping for the moment that $Irr(A)$ is an actual set, we have the map from Theorem 1, let’s start to prove now –

Firstly, each $End_A(W)$ is semi-simple as an $A$ -module being isomorphism to a direct sum of copies of $W$ (see the first section). Hence for any finite collection of irreducible $A$ -modules $W_1 . . . , W_r$ ,

$\prod_{1\leq i\leq r} End_{D_{W_i}}(W_i)$

is the isotypic decomposition of a semi-simple A-module. Secondly, we know that for each irreducible $W$ the projection $A \to End_{D_W}(W)$ is surjective (by Burnside’s theorem above). Hence the image of the map

$A \to \prod_{1\leq i\leq r}End_{D_{W_i}} (W_i) = \prod_{1\leq i\leq r} \underbrace{W_{i}\oplus... \oplus W_i}_{n_i}$

is a submodule (the last equality above follows from equation 2). We have already seen that every submodule of a semi-simple module is of the form

$\prod_{1\leq i\leq r}\underbrace{W_{i}\oplus... \oplus W_i}_{m_i}$

where $n_i \leq dim_{D_{W_i}}(W_i)$ , but since the map is induced by surjective maps $A\to End_{W_i}(W_i)$ it must intersect each $W_{i}$ non-trivially, hence $m_i = n_i$ and we are done.

The case of C*-algebras

The Jacobson radical

Given an arbitrary ring $R$ the Jacobson radical of $R$ , denoted $J(R)$ , is defined to be the subset of $R$ consisting of all elements which annihilates all irreducible (right) $R$ -modules. Equivalently it is the intersection in $R$ of all maximal ideals of $R$ . We will use both these definitions shortly.

The first definition is clearly interesting since it shows that the Jacobson radical is precisely the kernel of the map in Wedderburn’s theorem,

$A\to \prod_{w\in Irr(A)}M_{n_W}(D^{op}_W)$

Hence in the cases where $J(A) = 0$ we Wedderburn’s theorem produces an isomorphism of $A$ -modules. The good news is that for C*-algebras we have the following –

Theorme 2 $J(B) = 0$ for any C*-algebra $B$ .

Proof

The Gelfand-Naimark theorem produces a faithful representation $A\to \bigoplus_{\phi\in Pure(A)} \pi_\phi(A)$ as the direct sum of representations obtained using the GNS-construction on each pure state $\phi$ on $A$ . Recall that the GNS-representation of a state $\phi$ is irriducible if and only if the state is pure. Hence the intersections of all the kernels of the (topologically) irreducible representations $\pi_\phi$ where $\phi$ ranges over all pure states must be zero.
Now we use a corollary of Kaddison transitivity theorem which states that a representation of a C*-algebra $A$ is topologically irreducible if and only if it is algebraically irreducible. The claim now follows.

Classification of finite dimensional C*-algebras

Given a C*-algebra $A$ over $\mathbb{C}$ we can show it is isomorphic to a direct sum of full matrix algebras, so none of the finite Lie groups or non-unital matrix algebras are C*-algebras. Here we will rely on the machinery of the previous sections to prove the following theorem (see Theorem 6.3.8 of [1] for an alternative proof of this theorem) –

Theorem 3 Every finite-dimensional C*-algebra $A$ is of the form

$A = \bigoplus_{i=1, ..., m}M_{n_i}(\mathbb{C})$

Proof

Since $\mathbb{C}$ is algebraically closed we know that $D = \mathbb{C}$ , by the comment at the beginning of this post, hence Wedderburn’s theorem (Theorem 1) gives us a surjective morphism of algebras

$A\mapsto \bigoplus M_{n_i}(\mathbb{C})$

where $n_i$ is the dimension of some (isomorphism class of algebraically) irreducible representation and $\pi_i: A\to M_{n_i}(\mathbb{C})$ .
Theorem 2 now shows that this is indeed bijective, hence $A$ is algebraically isomorphic to $\bigoplus M_{n_i}(\mathbb{C})$ , and finally given it the $\star$ -operation inherited from $A$ yields a $\star$ -homomorphism.

The induced involution on the matrix algebras in the above proof can be assumed to be the usual conjugate-transpose after composing it with an automorphism given by conjugation by a self-adjoint invertible matrix (since all involutions of a matrix algebra are similarity transforms of each other). This would avoid relying on knowledge of the existence of a *-representation in every isomorphism class of (algebraically) irreducible representations of $A$ and the transitivity theorem.

It is likely that this post will grow in the future as I add more material regarding compact operators and other well behaved infinite $\star$ -algebras for which similar results can be extrapolated, but for now this will have to do.

Bibliography

[1] – G. J. Murphy C*-algebras and operator theory. Academic press, 2014.
[2] – G. Ellingsrud, – Lecture notes in representation theory – found here.

Wedderburn’s theorem and finite dimensional C*-algebras

Introduction

Some notation and a note on the endomorphism ring $End_A(V)$

Sem-simple $A$ -modules

Proof of Wedderburn’s theorem

The case of C*-algebras

The Jacobson radical

Classification of finite dimensional C*-algebras

Bibliography

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Introduction

Some notation and a note on the endomorphism ring

Sem-simple -modules

Proof of Wedderburn’s theorem

The case of C*-algebras

The Jacobson radical

Classification of finite dimensional C*-algebras

Bibliography

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Some notation and a note on the endomorphism ring $End_A(V)$

Sem-simple $A$ -modules