Math 202B: Lecture 15

Let V be a Hilbert space of dimension 1 < V < \infty and \mathcal{L}(V) be the algebra of all linear operators A \colon V \to V. As we saw in Lecture 14, the trace on \mathcal{L}(V) is, up to scaling, the unique linear functional \tau \colon \mathcal{L}(V) \to \mathbb{C} such that \tau(AB)=\tau(BA). Thus, the only linear functional satisfying \tau(I)=1 and \tau(AB)=\tau(BA) is

\tau(A) = \frac{1}{\dim V}\mathrm{Tr}(A), \quad A \in \mathcal{L}(V).

This fact allows us to prove a fact claimed early on in the course.

Problem 15.1. Prove that there does not exist an algebra homomorphism \chi \colon \mathcal{L}(V) \to \mathbb{C}.

The fact that \mathcal{L}(V) admits not even a single homomorphism to \mathbb{C} indicates that this is an algebra with the highest degree of noncommutativity. According to our definitions in Week One, the degree of noncommutativity of an algebra is formalized as the dimension of its center. We now prove that the center of \mathcal{L}(V) is indeed one-dimensional. The key notion is the following.

Definition 15.1. Given a subalgebra \mathcal{A} of \mathcal{L}(V), a subspace W of V is said to be \mathcal{A}–invariant if for every A \in \mathcal{A} and every w \in W we have Aw \in W.

Every subalgebra \mathcal{A} of \mathcal{L}(V) has at least two invariant subspaces, namely the zero space \{0_V\} and the total space V. An important theorem of Burnside characterizes subalgebras of \mathcal{L}(V) for which this minimum is achieved.

Theorem 15.1. (Burnside) A subalgebra \mathcal{A} of \mathcal{L}(V) has exactly two invariant subspaces if and only if \mathcal{A}=\mathcal{L}(V).

Proof: An elementary proof of Burnside’s theorem by induction in the dimension of V can be found here.

-QED

Corollary 15.2. A proper subalgebra \mathcal{A} of \mathcal{L}(V) has at least four invariant subspaces.

Proof: The zero space and the total space are \mathcal{A}-invariant, and since \mathcal{A} \neq \mathcal{L}(V) there is a third \mathcal{A}-invariant subspace \{0_V\} < W < \mathcal{L}(V), by Burnside’s theorem. The orthogonal complement W^\perp of W is again a nonzero proper subspace of V, and we claim it is \mathcal{A}-invariant. Indeed, let x \in W^\perp and A \in \mathcal{A}. Then, for any w \in W we have

\langle w,Ax \rangle = \langle A^*w,x \rangle = 0,

where we used the fact that W is \mathcal{A}-invariant. This shows Ax \in W^\perp, i.e. W^\perp is \mathcal{A}-invariant.

– QED

Using Burnside’s theorem we can characterize the center of \mathcal{L}(V).

Theorem 15.2. The center of \mathcal{L}(V) is the set \mathbb{C}I of scalar operators.

Proof: Let Z \in \mathcal{L}(V) be a central element, and let \zeta \in \mathbb{C} be an eigenvalue of Z. Then, the \zeta-eigenspace of Z is an \mathcal{L}(V)-invariant subspace, since if v \in V is an eigenvector of Z corresponding to \zeta and A \in \mathcal{L}(V) is any operator we have

ZAv = AZv =\lambda Av,

which shows that Av is again in the \zeta-eigenspace of Z. Since this eigenspace is not the zero space, Burnside’s theorem forces it to be all of V, i.e. Z=\zeta I.

-QED

We have succeeded in describing the center of \mathcal{L}(V) explicitly, and ideally we would like to have an explicit description of all subalgebras of \mathcal{L}(V). We were able to do this for function algebras \mathcal{F}(X), where subalgebras are indexed by partitions \mathsf{P} of the finite set X. The classification is very simple: the subalgebra \mathcal{A}(\mathsf{P}) of a given partition \mathsf{P} of X is the set of all functions on X which are constant on the blocks of \mathsf{P}. A nice feature of this classification is that it shows every subalgebra of \mathcal{F}(X) is itself isomorphic to a function algebra, because \mathcal{A}(\mathsf{P}) is isomorphic to \mathcal{F}(\mathsf{P}).

We can attempt to classify subalgebras of \mathcal{L}(V) by adapting our approach to the classification of subalgebras of \mathcal{F}(X). The first step is to formulate the vector space version of partitions of a set.

Definition 15.2. A linear partition of V is a set \mathsf{W} of nonzero subspaces of V whose union spans V. We refer to the subspaces W \in \mathsf{W} as the blocks of \mathsf{W}.

Given a linear partition \mathsf{W} of V, it is not useful to consider the set of linear operators in \mathcal{L}(V) constant on the blocks of \mathsf{W}. Indeed, there is only one such operator, namely the zero operator. Rather, we define \mathcal{A}(\mathsf{W}) to be the set of all operators A \in \mathcal{L}(V) such that every block W \in \mathsf{W} is A-invariant.

Problem 15.2. Prove that A(\mathsf{W}) is a subalgebra of \mathcal{L}(V).

Here our construction starts to diverge from what we saw in the setting of function algebras: unless \mathsf{W}=\{V\} is the linear partition of V consisting of a single block, in which case \mathcal{A}(\mathsf{W})=\mathcal{L}(V), the subalgebra \mathcal{A}(\mathsf{W}) is definitely not isomorphic to the algebra of all linear operators on a Hilbert space. Indeed, consider the case where \mathsf{W}=\{W_1,W_2\} is a linear partition of V with two blocks. Let e_1,\dots,e_m be an ordered basis of W_1, and let f_1,\dots,f_n be an ordered basis of W_2. Then e_1,\dots,e_m,f_1,\dots,f_n is an ordered basis of V, and \mathcal{A}(\mathsf{W}) consists of all operators in A\in \mathcal{L}(V) whose matrix relative to this ordered basis has the form

[A]_{(e_1,\dots,e_m,f_1,\dots,f_n)}=\begin{bmatrix} M_1 & {} \\ {} & M_2 \end{bmatrix},

with M_1 \in \mathbb{C}^{m \times m} and M_2 \in \mathbb{C}^{n\times n}. Thus, \mathcal{A}(\{W_1,W_2\}) is not isomorphic to a linear algebra, but rather to a direct sum of linear algebras,

\mathcal{A}(\{W_1,W_2\}) = \mathcal{L}(W_1) \oplus \mathcal{L}(W_2).

The question now is whether the above can reversed: given a subalgebra \mathcal{A} of \mathcal{L}(V), does there exist a linear partition \mathsf{W} such that \mathcal{A}=\mathcal{A}(\mathsf{W})? If \mathcal{A}=\mathcal{L}(V), the answer is clearly “yes” since we have \mathcal{L}(V) = \mathcal{A}(\{V\}). Thus we consider the case where \mathcal{A} is a proper subalgebra of \mathcal{L}(V). Our starting point is Burnside’s theorem, which guarantees the existence of an \mathcal{A}-invariant subspace \{0_V\} < W < V. Together with the corollary to Burnside’s theorem, we thus have a linear partition \mathsf{W} = \{W,W^\perp\} of V whose blocks are \mathcal{A}-invariant. We now ask whether W has a proper non-trivial subspace invariant under \mathcal{A}. If it does we can split it into the orthogonal direct sum of two smaller \mathcal{A}-invariant subspaces; if not W is said to be \mathcal{A}-irreducible. Continuing this process, for both W and W^\perp, we get the following.

Theorem 15.4. (Maschke’s theorem) There exists a linear partition \mathsf{W} of V whose blocks are \mathcal{A}-invariant and \mathcal{A}-irreducible subspaces.

Now consider the linear partition \mathsf{W} of V into \mathcal{A}-invariant, \mathcal{A}-irreducible subspaces we have constructed. It is tempting to hope that \mathcal{A} = \mathcal{A}(\mathsf{W}). This is almost correct. Let us say that two blocks W,W' \in \mathsf{W} are \mathcal{A}–equivalent if it is possible to choose a basis in W and a basis in W' such that the matrix of every A \in \mathcal{A}, viewed as an operator on W, coincides with the matrix of A viewed as an operator on W'. This is an equivalence relation on the blocks of \mathsf{W}. Let \Lambda be a set parameterizing the corresponding equivalence classes of blocks, and for each \lambda \in \Lambda choose a representative V^\lambda of the corresponding equivalence class.

Theorem 15.5. (Classification Theorem for subalgebras of linear algebras) The subalgebra \mathcal{A} is isomorphic to \bigoplus_{\lambda \in \Lambda} \mathcal{L}(V^\lambda).

We are not going to prove the above classification in full right now – in the coming lectures we will prove a special case of it for homomorphic images of convolution algebras of finite groups. If time permits, we will prove the full theorem later.

Published by Jonathan Novak

Mathematician at UC San Diego.

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