Math 202B: Lecture 15

In this lecture we will review the abstract theory of algebras.

Algebras. We began with the general definition of an algebra: a finite-dimensional complex vector space \mathcal{A} together with a multiplication (A,B) \mapsto AB and a conjugation A \mapsto A^* satisfying certain natural axioms. These axioms are to a large extent inspired by the key example \mathcal{A}=\mathrm{End}(V), the endomorphism algebra of a finite-dimensional Hilbert space V, which was the sole subject of Math 202A. We use much of the same terminology for general algebras:

  • Normal: A and A^* commute.
  • Unitary: U is invertible with inverse U^*.
  • Selfadjoint: X^*=X. Important subtypes:
    • Idempotent: satisfies E^2=E.
    • Positive: factors as P=A^*A.

We obtained a simple classification of commutative algebras.

Theorem: \mathcal{A} is commutative if and only if all its elements are normal.

Subalgebras. We next had a general discussion of subalgebras: subspaces \mathcal{B} of an algebra \mathcal{A} which are themselves algebras. More precisely, \mathcal{B} is a vector subspace of \mathcal{A} which is closed under multiplication and conjugation, and contains the multiplicative identity I. In particular, the zero space is not a subalgebra, and \mathbb{C}I is the smallest subalgebra of \mathcal{A} in the sense that it is contained in every subalgebra. We discussed the lattice of subalgebras and for any set S \subseteq \mathcal{A} defined the algebra \mathrm{Alg}(S) it generates to be the intersection of all subalgebras containing S.

A particularly important subalgebra is the center Z(\mathcal{A}), the set of elements in \mathcal{A} which commute with every other element. The dimension of Z(\mathcal{A})$ is a measure of how commutative \mathcal{A} is. Whenever we encounter a particular algebra we wish to understand, our first step should be to understand its center and compute its dimension.

Subalgebras always come in pairs: if \mathcal{B} is a subalgebra of \mathcal{A}, there is an affiliated subalgebra Z(\mathcal{B}), the centralizer of \mathcal{B}, the set of elements in \mathcal{A} commuting with every element of B. This concept also helps us to identity maximal abelian subalgebras in \mathcal{A}.

Theorem: An abelian subalgebra \mathcal{B} \leq \mathcal{A} is a MASA if and only if Z(\mathcal{B})=\mathcal{B}.

Ideals. We did not spend much time on ideals because we don’t really need to know much about them for our purposes. However, this concept arises naturally from the fact that, given an algebra homomorphism \Phi \colon \mathcal{A} \to \mathcal{B}, the image of \Phi is a subalgebra of \mathcal{B} but the kernel of \Phi is not generally a subalgebra of \mathcal{A}.. However, \mathrm{Ker}(\Phi) is a subspace of \mathcal{A} closed under conjugation with a special absorption property: if J \in \mathrm{Ker}(\Phi), then both AJ and JA are contained in \mathrm{Ker}(\Phi) for all A \in \mathcal{A}. Any *-closed subspace \mathcal{J} of \mathcal{A} with this property is called an ideal and the quotient vector space \mathcal{A}/\mathcal{J} whose points are sets of the form

[A]_\mathcal{J} = \{A+J \colon J \in \mathcal{J}\}

has a natural algebra structure given by

[A]_\mathcal{J}[B]_\mathcal{J}=[AB]_\mathcal{J} \quad\text{and}\quad [A]_\mathcal{J}^*= [A^*]_\mathcal{J},

with the multiplicative identity being [I]_\mathcal{J}. The most basic theorem here is the First Isomorphism Theorem for the category of algebras.

Theorem: For any algebra homomorphism \Phi \colon \mathcal{A} \to \mathcal{B} the algebras \mathcal{A}/\mathrm{Ker}(\Phi) and \Phi(\mathcal{A}) are isomorphic via [A]_{\mathrm{Ker}(\Phi)}\mapsto \Phi(A).

von Neumann algebras. We arrived at this special class of algebras by wondering whether we can equip a given algebra \mathcal{A} with a scalar product that is compatible with its algebra structure in the sense that the Frobenius identities

\langle AB,C \rangle = \langle B,A^*C\rangle \quad\text{and}\quad \langle AB,C\rangle = A,CB^*\rangle

hold for all A,B,C \in \mathcal{A}. When such a scalar product exists and further satisfies \langle I,I \rangle =1, we refer to it as a Frobenius scalar product. We defined a von Neumann algebra to be a pair (\mathcal{A},\langle \cdot,\cdot\rangle) consisting of an algebra equipped with a Frobenius scalar product.

We reduced the search for Frobenius scalar products on \mathcal{A} to the search for linear functionals with certain special properties. First, we established that the left Frobenius identity holds for a given scalar product if and only if it is of the form

\langle A,B \rangle = \tau(A^*B)

with \tau \colon \mathcal{A} \to \mathbb{C} a linear functional satisfying \tau(A^*A) \geq 0 for all A \in \mathcal{A}, with equality if and only if A=0_\mathcal{A}. If such a \tau does exist then it is unique, and the condition \langle I,I \rangle =1 is equivalent to \tau(I)=1. We thus defined states to be linear functionals on \mathcal{A} satisfying

\tau(I)=1 \quad\text{and}\quad \tau(A^*A) \geq 0,

and resolved to call a state faithful if \tau(A^*A) \geq 0 forces A=0_\mathcal{A}. We thus conclude that scalar products on \mathcal{A} satisfying the left Frobenius identity and the normalization condition \langle I,I \rangle =1 are in bijection with faithful states on \mathcal{A}. Moreover, we the scalar product \langle A,B\rangle=\tau(A^*B) defined by a faithful state \tau satisfies the right Frobenius identity if and only if \tau is a trace, meaning that \tau(AB)=\tau(BA) for all A,B \in \mathcal{A}. Hence we may equivalently define a von Neumann algebra to be a pair (A,\tau) consisting of an algebra together with a faithful tracial state.

Summary. The above is essentially the abstract theory of algebras that we have developed so far. The rest of the course content concerns particular algebras: function algebras, endomorphism algebras, convolution algebras, and class algebras. A review of these concrete algebras is up next.

Published by Jonathan Novak

Mathematician at UC San Diego.

Leave a Reply