Von Neumann Algebras

A von Neumann algebra is a unital star subalgebra of bounded operators closed under weak operator limits. It supplies the measurable projections and monotone limits needed for spectral events and normal states. Unlike an abstract C*-norm completion, this closure is taken in a specified Hilbert-space representation; different representations of one C*-algebra can produce different von Neumann algebras.

Required background. C*-Algebras explains norm closure. Tracking Topologies distinguishes norm, strong, and weak operator convergence.

Weak closure depends on the represented vectors

Let $\mathcal M\subset\mathcal B(\mathcal H)$ contain $I$ and be closed under products, sums, scalar multiples, and adjoints. Weak operator convergence means

$$A_\alpha\longrightarrow A \quad\Longleftrightarrow\quad \langle\xi,A_\alpha\eta\rangle \longrightarrow\langle\xi,A\eta\rangle \quad(\xi,\eta\in\mathcal H).$$

If every such limit of elements of $\mathcal M$ remains in $\mathcal M$, then $\mathcal M$ is a von Neumann algebra. By the bicommutant theorem, this is equivalent to strong operator closedness and to $\mathcal M=\mathcal M''$. The theorem requires the unital star-algebra hypotheses; weak and strong closure need not coincide for arbitrary sets.

Every von Neumann algebra is norm closed and therefore a C*-algebra. The converse fails. For an infinite-dimensional separable $\mathcal H$, finite-rank projections $P_n\uparrow I$ converge strongly, but $I$ is not compact. Thus $\mathcal K(\mathcal H)$ is norm closed and not strongly closed.

Even adjoining the identity does not fix this example: $\mathcal K(\mathcal H)+\mathbb CI$ is strongly dense in $\mathcal B(\mathcal H)$ but is still a proper norm-closed subalgebra.

Measurable functions enter after taking weak closure

Represent $C([0,1])$ on $L^2([0,1],dx)$ by multiplication. This representation is faithful because Lebesgue measure has full support: a nonzero continuous function cannot vanish almost everywhere.

Its von Neumann closure is the algebra of multiplication operators by $L^\infty([0,1],dx)$ functions. To see why new operators enter, approximate $1_{[0,1/2]}$ almost everywhere by uniformly bounded continuous functions $f_n$. Dominated convergence gives

$$\|(M_{f_n}-M_{1_{[0,1/2]}})\psi\|_2\longrightarrow0 \qquad(\psi\in L^2).$$

This is strong convergence, although there is no uniform approximation of the jump by continuous functions. General bounded measurable multipliers are obtained by similar approximation and closure; the identification also follows from the multiplication algebra’s commutant.

By contrast, representing the same $C([0,1])$ through evaluation at one point gives only $\mathbb C$ on a one-dimensional Hilbert space. Neither closure is a representation-independent operation on the abstract algebra alone.

Spectral projections and affiliated observables

If $A=A^*\in\mathcal M$, its bounded Borel functional calculus lies in $\mathcal M$. In particular, every spectral projection $E_A(\Delta)$ belongs to the algebra. Continuous approximants and monotone limits connect this fact to weak closure.

An unbounded self-adjoint operator $T$ is not an element of $\mathcal M$. It is affiliated with $\mathcal M$ when all its spectral projections belong to $\mathcal M$. Equivalently, its spectral calculus commutes with every unitary in $\mathcal M'$, with the corresponding domain invariance. This is a spectral statement, stronger than a formal commutator on a convenient domain.

For example, an unbounded measurable real function $f$ gives an affiliated multiplication operator on $L^2([0,1])$. Its domain is still

$$D(M_f)=\{\psi\in L^2:f\psi\in L^2\}.$$

Affiliation organizes observable membership; it does not eliminate domain or moment conditions. The spectral probability distribution exists in a normal state even when its first or second moment diverges.

Normal functionals form the predual

A von Neumann algebra has a canonical Banach predual $\mathcal M_*$, whose positive normalized elements are its normal states. The corresponding weak-star topology on $\mathcal M=(\mathcal M_*)^*$ is called ultraweak.

For $\mathcal B(\mathcal H)$, the predual is the trace-class operators, paired by $\operatorname{Tr}(TA)$. For a concrete subalgebra, normal functionals are restrictions of such functionals; the representing trace-class operator on the ambient Hilbert space need not be unique or belong to the subalgebra.

On uniformly norm-bounded sets, weak operator and ultraweak topologies coincide in a concrete von Neumann algebra. Without that bound, the finite matrix-element tests and the trace-class tests should not be interchanged. Normal and Singular States develops the state-side distinction.

Exercise: a factor can act reducibly

Let $\mathcal M=M_n(\mathbb C)\otimes I_m$ on $\mathbb C^n\otimes\mathbb C^m$, with $m>1$. Find its center and explain whether it acts irreducibly.

Solution

Its center is $\mathbb CI$: a matrix commuting with every element of $M_n$ is scalar. Thus $\mathcal M$ is a factor, meaning that its center is trivial. Its commutant nevertheless contains $I_n\otimes M_m$, including nontrivial projections, so its action is reducible. Being a factor concerns the algebra’s center; irreducibility concerns its entire commutant in the chosen representation.

References

• Bratteli, Ola, and Derek W. Robinson, Operator Algebras and Quantum Statistical Mechanics 1, 2nd ed., Springer, 1987, doi:10.1007/978-3-662-02520-8 — weak closure, preduals, and normal functionals.
• Jones, Vaughan F. R., Von Neumann Algebras, lecture notes, 2010, author’s notes, Chapters 3–5 — operator topologies, the bicommutant theorem, and examples.