Math 202: Endomorphism Algebras

Math 202A was an expression of the belief that the quantization of a finite set X is the Hilbert space V=\mathcal{F}(X) containing X as an orthonormal basis. Math 202B extends the gospel from Hilbert spaces to algebras, taking \mathcal{F}(X) as the prototypical commutative algebra and \mathrm{End}(V)=\mathrm{End}\mathcal{F}(X) as the prototypical noncommutative algebra.

The endomorphism algebra \mathrm{End}(V)=\mathrm{End}\mathcal{F}(X) contains a subalgebra isomorphic to \mathcal{F}(X), namely the diagonal subalgebra \mathcal{D}(X) consisting of operators on V such that

Ax=\widehat{A}(x)x, \quad x \in X,

where \widehat{A}(x) \in \mathbb{C} is a scalar. Indeed, the isomorphism

\Phi \colon \mathcal{D}(X) \longrightarrow \mathcal{F}(X)

defined by \Phi(A)=\widehat{A} is the Fourier transform on \mathcal{D}(X), and it can equivalently be described in terms of a basis of \mathrm{End}(V) constructed from the orthonormal basis X \subset V, namely the basis of elementary operators

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

These operators multiply as

E_{xy}E_{wz}=E_{xz}\langle y,w\rangle, \quad x,y,z,w \in X,

and conjugate as

E_{xy}^*=E_{yx}, \quad x,y \in X,

making \{E_{xx} \colon x \in X\} a set of pairwise orthogonal projections that spans the diagonal subalgebra \mathcal{D}(X). The Fourier transform \Phi \colon \mathcal{D}(X) \to \mathcal{F}(X) is \Phi(E_{xx})=E_x.

The algebra \mathrm{End}(V)=\mathrm{End}\mathcal{F}(X) is bigger and harder to understand than \mathcal{F}(X), which is why the algebra of endomorphisms on a finite-dimensional Hilbert space is the subject of entire courses and the algebra of functions on a finite set is not. The first problem treated is generally: find necessary and sufficient conditions on an operator A \in \mathrm{End}(V) such that A \in \mathcal{D}(Y) for some orthonormal basis Y \subset V. Typically, this is posed as a problem about matrices: you are given the matrix of A relative to the computational basis X \subset V, and almost certainly the entity who gave it to you this matrix ensured it was not diagonal, so A \not\in \mathcal{D}(X).

Theorem E1. There exists and orthonormal basis Y \subset V such that A \in \mathcal{D}(Y) if and only if A is normal.

Consequently, if A is normal then it lives in \mathcal{D}(Y) and its Fourier transform \widehat{A} \in \mathcal{F}(Y) outputs eigenvalues, Ay=\widehat{A}(y)y. Our general taxonomy of special elements in an abstract algebra gives you a number of conditions which imply that an operator is normal.

The diagonalization problem puts a particular focus on diagonal subalgebras of \mathrm{End}(V), and hence it is useful to characterize these objects from a second point of view.

Theorem E2. A commutative subalgebra \mathcal{A} \leq \mathrm{End}(V) is a maximal commutative subalgebra if and only if \mathcal{A}=\mathcal{D}(Y) for some orthonormal basis Y \subset V.

So the diagonal subalgebras of \mathrm{End}(V) are exactly the maximal elements of the poset of commutative subalgebras of \mathrm{End}(V). The natural question is then whether one has a Fourier transform defined on every commutative subalgebra of \mathrm{End}(V) which converts it into a function algebra.

Theorem E3. A subalgebra \mathcal{A} \leq \mathrm{End}(V) is commutative if and only if it is isomorphic to a subalgebra of \mathcal{F}(Y) for some orthonormal basis Y \subset V.

The philistines call this result “simultaneous diagonalization,” because the non-obvious direction is proved as follows: take a basis A_1,\dots,A_m of \mathcal{A}, use the general fact that an algebra is commutative if and only if all its elements are normal, and using Theorem E1 show that there exists an orthonormal basis Y \subset V such that A_1,\dots,A_m \in \mathcal{D}(Y). Since all subalgebras of function algebras are function algebras, Theorem E3 proves that all commutative subalgebras of \mathrm{End}(V) are function algebras.

The classification of noncommutative subalgebras of \mathrm{End}(V) is more complicated, and this is precisely the point where linear algebra starts to look like representation theory: in order to understand noncommutative subalgebras \mathcal{A} \leq \mathrm{End}(V) it is no longer sufficient to look for one-dimensional \mathcal{A}-invariant subspaces of V. A fundamental result due to Burnside (sometimes called the fundamental theorem of noncommutative algebra) asserts that every subalgebra \mathcal{A} of \mathrm{End}(V) except \mathrm{End}(V) itself admits a proper nontrivial invariant subspace.

Theorem E4. We have \mathcal{A}=\mathrm{End}(V) if and only if \mathcal{A} has exactly two invariant subspaces. In fact, if \mathcal{A} is a proper subalgebra of \mathrm{End}(V), it has at least four invariant subspaces.

The first part of Theorem E4 is Burnside’s theorem, the second is Maschke’s theorem: W is \mathcal{A}-invariant if and only if W^\perp is.

Given a subalgebra \mathcal{A} of \mathrm{End}(V), it is helpful to order the set of all \mathcal{A}-invariant subspaces by inclusion. This forms an induced sublattice of the lattice of all subspaces of V with maximal element V. Nonzero elements W of the lattice of \mathcal{A}-invariant subspaces are called representations of \mathcal{A} because restricting every A \in \mathcal{A} to W gives an algebra homomomorphism \rho \colon \mathcal{A} \to \mathrm{End}(W). A representation of \mathcal{A} is called irreducible if the only element below it in the lattice of \mathcal{A}-invariant subspaces is \{0_V\}.

Theorem E5. For every \mathcal{A} \leq \mathrm{End}(V), there exists an orthogonal decomposition of V into irreducible representations of \mathcal{A}.

Note that Theorem E5 is a noncommutative generalization of Theorem E3, which gives an orthogonal decomposition of V into one-dimensional invariant subspaces of a commutative subalgebra \mathcal{A} \leq \mathrm{End}(V). However, while Theorem E3 is enough to classify commutative subalgebras of \mathrm{End}(V), Theorem E5 is not quite enough to classify all subalgebras of \mathrm{End}(V), and you are not expected to know the classification of subalgebras of \mathrm{End}(V).

We did however classify states and traces on \mathrm{End}(V)=\mathrm{End}\mathcal{F}(X). First, we proved that the function defined by

\mathrm{Tr}_X(A) = \sum\limits_{x \in X} \langle x,Ax\rangle, \quad A \in \mathrm{End}(V),$

is a trace on \mathrm{End}(V). Second, we proved that if Y \subset V is any orthonormal basis, the linear functional

\mathrm{Tr}_Y(A)\sum\limits_{y \in Y} \langle y,Ay\rangle

coincides with \mathrm{Tr}_X. Therefore, we simply denote by \mathrm{Tr} the function defined by \mathrm{Tr}_X.

Theorem E6. Up to scaling, \mathrm{Tr} is the unique trace on \mathrm{End}(V).

The classification of states on \mathrm{End}(V) is more flexible but still very structured. Recall from Part \mathcal{A} that an element P of an abstract algebra \mathcal{A} is said to be nonnegative if it can be factored in the form P=A^*A. In the particular case of the endomorphism algebra \mathrm{End}(V), a nonngegative operator P \in \mathrm{End}(V) is called a density operator if it satisfies \mathrm{Tr}(P)=1. Note that this condition is equivalent to

\mathrm{Tr} P =\sum\limits_{x \in X} \langle x,Px\rangle= \sum\limits_{x \in X} \langle Ax,Ax\rangle = 1,

which is equivalent to saying that the function in \mathcal{F}(X) defined by x \mapsto \langle x,Px\rangle is a probability function in \mathcal{F}(X), which makes density operators a natural noncommutative analogue of probability functions.

Theorem E7. For every density operator P, the linear functional \sigma(A)=\mathrm{Tr}(PA) is a state on \mathrm{End}(V), and every state on \mathrm{End}(V) is of this form. Moreover, \sigma is faithful if and only if \mathrm{rank}(P) = \dim V.

The significance of faithful states is that they allow us to define a noncommutative analogue of the Fourier transform that maps abstract algebras to subalgebras of endomorphism algebras. Namely, let \mathcal{A} be an abstract algebra, and suppose that \sigma is a faithful state on \mathcal{A} — like the situation with a Fourier basis, it is not necessarily the case that a faithful state exists, but suppose one does. Then, we have a left-Frobenius scalar product on \mathcal{A} defined by \langle A,B \rangle=\sigma(A^*B), so that we can view \mathcal{A} as a Hilbert space and consider the corresponding endomorphism algebra \mathrm{End}(\mathcal{A}).

Theorem E8. The function \Psi \colon \mathcal{A} \longrightarrow \mathrm{End}(\mathcal{A}) defined by \Psi(A)B=AB, A,B \in \mathcal{A}, is an injective algebra homomorphism.

The map \Psi in Theorem E8 is called the Wedderburn transform of \mathcal{A} relative to \sigma. Theorem 8 says that any algebra which supports a faithful state can be transformed into a subalgebra of an endomorphism algebra using the corresponding Wedderburn transform. This should be compared with the statement that any algebra which supports a Fourier basis can be transformed into a function algebra using the Fourier transform. One consequence of the Wedderburn transform is the following.

Theorem E9. An algebra \mathcal{A} supports a faithful tracial state if and only if it is isomorphic to a subalgebra of an endomorphism algebra.

There is a fundamental relationship between the Fourier transform and the Wedderburn transform which is extremely important in situations where both can be implemented. Let \mathcal{A} be an algebra which admits a Fourier basis F=\{F^\lambda \colon \lambda \in \Lambda\}. Then, the scalar product on \mathcal{A} in which the Fourier basis is orthonormal is a Frobenius scalar product on \mathcal{A}. This means that the Wedderburn transform \Psi \colon \mathcal{A} \to \mathrm{End}(\mathcal{A}) with respect to this Frobenius scalar product is defined.

Theorem E10. For every A \in \mathcal{A}, we have \Psi(A) \in \mathcal{D}(F), and moreover \Psi(A)F^\lambda = \widehat{A}(\lambda)F^\lambda for all \lambda \in \Lambda.

Published by Jonathan Novak

Mathematician at UC San Diego.

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