Week 2 — Von Neumann Algebras and the Double Commutant

We strengthened “Banach algebra” to “C*-algebra” by adding one axiom (the C*-identity) and discovered enormous structural rigidity. We now strengthen “C*-algebra” to “von Neumann algebra” by changing the topology in which we close. The price is a concrete definition, stated inside a chosen representation; §6 explains why nothing essential depends on that choice. The benefit: a wealth of projections, a topology in which the bicommutant theorem trivializes the algebra-vs-topology distinction, and the foundation for everything in modular theory and beyond.

How to use this chapter

• In class: compare norm, SOT, and WOT on explicit operator sequences before stating the bicommutant theorem; then use commutants to introduce factors and normality.
• For self-study: calculate the commutants of the diagonal algebra and of $\mathcal{B}({\mathcal{H}}_1)\otimes 1$. Track exactly where boundedness, nets rather than sequences, and the distinction between WOT and SOT enter.
• Instructor checkpoint: test the two equivalences that students most often blur: irreducibility means commutant $=\mathbb{C}1$, while a unital $\ast$-algebra has the same WOT and SOT closures, not the same two topologies.

0. Reading

Primary: Bratteli & Robinson Vol. I, §2.4 (von Neumann algebras and the bicommutant theorem).

Secondary:

• Murphy, C*-algebras and Operator Theory, ch. 4.
• Takesaki Vol. I, ch. II–III.
• Sakai, C*-algebras and W*-algebras, ch. 1 (for the abstract characterization of vN algebras).

Reference, used in passing: Connes, Noncommutative Geometry, ch. V.

Proof-status legend

This note uses the standard course proof-status labels (codified in the note-quality template and conventions.md): [Proved.] / [Sketched.] / [Stated only — refs.] / [Example-only.] / [Hypothesis-explicit.].

1. Why a new closure

The C*-algebras of Week 1 are abstract: they can be defined without reference to any Hilbert space. Once we fix a state $\omega$ and run the GNS construction, we obtain a concrete realization ${\pi }_{\omega }(\mathcal{A})\subset \mathcal{B}({\mathcal{H}}_{\omega })$. But this realization is a *-subalgebra closed in the norm topology — not in any of the weaker operator topologies.

Why does this matter? Three reasons.

(i) Spectral projections are missing. Suppose $a$ is a self-adjoint element of a C*-algebra $\mathcal{A}$ and $E\subset \mathbb{R}$ is a Borel set. The spectral projection $1_E(a)$ — the projection onto the spectral subspace where $a$ takes values in $E$ — is in general not in $\mathcal{A}$. It is only the limit, in the strong operator topology, of polynomials in $a$. Concretely: in $\mathcal{A}=C([0,1])$, the polynomial functions in $a(x)=x$ are dense in $C([0,1])$ in sup-norm, but the indicator functions $1_{[0,1/2]}$ are not in $C([0,1])$. We need a coarser topology — one that allows pointwise limits of bounded sequences of polynomials — to recover them.

Without spectral projections we cannot define yes/no measurements, decompose states into pure components, or develop the basic measurement theory of operator algebras.

(ii) Cyclic representations need not be irreducible. A cyclic vector only says that ${\pi }_{\omega }(\mathcal{A}){\Omega }_{\omega }$ is dense; it does not exclude invariant subspaces or a nontrivial commutant. By contrast, irreducibility is equivalent to

$${\pi }_{\omega }(\mathcal{A}{)}^{\prime }=\mathbb{C}1,$$

so an irreducible representation has no non-scalar commuting operators at all. Passing to the von Neumann closure makes the actual commutant and the associated multiplicities available as part of the operator-algebraic structure.

(iii) Many constructions are intrinsically weakly closed. In QFT we will eventually want to consider the algebra generated by Weyl operators with test functions in a region (Block C). This generation is a topological closure operation, and spectral projections and bounded Borel functions enter naturally. WOT and SOT are different topologies—even on norm-bounded sets, as Example 2.2 shows—but the bicommutant theorem says that the WOT- and SOT-closures of a unital $\ast$-subalgebra coincide. That equality of algebra closures, not an equality of topologies, is what we use.

The remedy is to enlarge ${\pi }_{\omega }(\mathcal{A})$ to a larger algebra by closing in a weaker topology. We close in the weak operator topology. Theorem 4.1 shows that the strong and σ-weak topologies produce exactly the same closed *-algebras, and the remark after Definition 3.1 explains why the definition is stated with the weak one.

2. Operator topologies on $\mathcal{B}(\mathcal{H})$

Recall that $\mathcal{B}(\mathcal{H})$ admits at least four useful topologies:

1. Norm topology: $T_n\to T$ if $\Vert T_n-T\Vert \to 0$.
2. Strong operator topology (SOT): $T_n\to T$ if $T_n\xi \to T\xi$ in norm for every $\xi \in \mathcal{H}$.
3. Weak operator topology (WOT): $T_n\to T$ if $⟨\eta ,T_n\xi ⟩\to ⟨\eta ,T\xi ⟩$ for every $\xi ,\eta \in \mathcal{H}$.
4. σ-weak topology (also called ultraweak): $T_n\to T$ if ${\sum }_i⟨{\eta }_i,T_n{\xi }_i⟩\to {\sum }_i⟨{\eta }_i,T{\xi }_i⟩$ for every pair of sequences $\{{\xi }_i\},\{{\eta }_i\}$ with $\sum \Vert {\xi }_i{\Vert }^2<\infty ,\sum \Vert {\eta }_i{\Vert }^2<\infty$.

Norm convergence implies SOT, which implies WOT. The σ-weak topology is in general finer than WOT, although the two induce the same topology on norm-bounded subsets of $\mathcal{B}(\mathcal{H})$. This last statement should not be confused with SOT versus WOT: those remain distinct on bounded sets.

Note that each condition above is written for sequences, which is how one computes with these topologies. The topologies themselves are defined by the corresponding seminorms, $T\mapsto \Vert T\xi \Vert$ for SOT, $T\mapsto |⟨\eta ,T\xi ⟩|$ for WOT, and $T\mapsto |{\sum }_i⟨{\eta }_i,T{\xi }_i⟩|$ for σ-weak, and limits and closures in them must in general be taken along nets. Sequences cannot even separate WOT from σ-weak. By the uniform boundedness principle a WOT-convergent sequence is norm-bounded, and on norm-bounded sets the two topologies agree, so the difference between them is carried entirely by unbounded nets (Problem 8*).

Physical picture. Each topology answers a different operational question about when two operators are “close.” Norm closeness is uniform closeness: for a self-adjoint difference $x=x^{\ast }$, $\Vert x\Vert ={\sup }_{\omega }|\omega (x)|$, so norm convergence means that the worst-case difference in expectation values over all states goes to zero. For a general operator the norm is instead recovered from matrix elements, or from states applied to $x^{\ast }x$. That is an extremely demanding notion — no experiment ever probes all states at once. WOT closeness is closeness of matrix elements $⟨\eta ,a_n\xi ⟩$, i.e. of transition amplitudes between specific states: $a_n\to a$ in WOT means every fixed matrix element converges. SOT is intermediate: the action on each fixed state vector converges. The physically useful closure is the weak one: it includes limits detected by all matrix elements in the chosen representation. That is precisely why spectral projections (yes/no questions about an observable) belong to the vN closure even when they are missing from the C*-algebra.

Lemma 2.1. [Proved.] In finite dimensions ($\dim \mathcal{H}=n<\infty$), all four topologies coincide.

Proof. $\mathcal{B}(\mathcal{H})\cong M_n(\mathbb{C})$ is finite-dimensional. Any two norms on a finite-dimensional vector space are equivalent (a standard linear-algebra result). The SOT and WOT are induced by families of seminorms, and on a finite-dimensional space they all reduce to the same norm topology. $\square$

In infinite dimensions, the topologies are genuinely distinct. The key examples:

Example 2.2 (Distinct topologies in $\mathcal{B}({\ell }^2)$). Let $S$ be the unilateral shift on ${\ell }^2$: $S{e}_n=e_{n+1}$. Consider the sequence $S^n$.

• Norm: $\Vert S^n\Vert =1$ for all $n$, so $S^n$ does not converge in norm.
• SOT: $S^n\xi \to 0$ for every $\xi \in {\ell }^2$? The component of $\xi$ at index $k$ is shifted to index $k+n$, which goes off to infinity. The norm of $S^n\xi$ remains $\Vert \xi \Vert$. So $S^n\xi$ does not converge to anything in norm — the SOT-limit doesn’t exist.
• However, $⟨\eta ,S^n\xi ⟩={\sum }_k\overline{{\eta }_{k+n}}{\xi }_k\to 0$ as $n\to \infty$ by Cauchy–Schwarz on the tail of $\eta$. So $S^n\to 0$ in WOT but not in SOT.

Lemma 2.3. [Sketched — separate continuity proved; non-joint via WOT counterexample.] On $\mathcal{B}(\mathcal{H})$, multiplication is jointly continuous in the norm topology, but only separately continuous in SOT and WOT in general.

WOT counterexample to joint continuity. Let $S$ be the unilateral shift on ${\ell }^2$. As in Example 2.2, $S^n\to 0$ and $S^{\ast n}\to 0$ in WOT (the matrix elements $⟨e_j,S^n{e}_k⟩$ vanish for $n$ large enough). Yet

$$S^{\ast n}\cdot S^n=1\text{for every }n\ge 0,$$

since $S^{\ast }$ is a left inverse of $S$. So $A_n:=S^{\ast n}$ and $B_n:=S^n$ satisfy $A_n\to 0$, $B_n\to 0$ in WOT, but $A_n{B}_n=1\nrightarrow 0=0\cdot 0$. Joint WOT-continuity of multiplication therefore fails.

(Note on SOT. In SOT, multiplication actually is jointly continuous on bounded sets — this follows from $\Vert (A_n{B}_n-AB)\xi \Vert \le \Vert A_n\Vert \Vert (B_n-B)\xi \Vert +\Vert (A_n-A)(B\xi )\Vert$ when $\Vert A_n\Vert$ is bounded. SOT joint continuity fails only on unbounded sets, where the counterexamples are more delicate. For this course the WOT failure above is the operationally relevant one.)

Proof of separate continuity. Suppose $T_n\to T$ in WOT and $S\in \mathcal{B}(\mathcal{H})$ is fixed. Then for any $\xi ,\eta$, $⟨\eta ,T_{n}S\xi ⟩=⟨\eta ,T_n(S\xi )⟩\to ⟨\eta ,T(S\xi )⟩=⟨\eta ,TS\xi ⟩,$ so $T_{n}S\to TS$ in WOT. The other side is similar (use $T_n^{\ast }\to T^{\ast }$ in WOT, which follows from the definition since $⟨\xi ,T_n^{\ast }\eta ⟩=\overline{⟨\eta ,T_n\xi ⟩}$). $\square$

The non-joint continuity of multiplication is the technical reason vN algebra theory requires care: products of converging sequences need not converge, and many “obvious” identity manipulations fail. This is why the bicommutant theorem (which sidesteps this issue) is so important.

3. Von Neumann algebras: definition

Definition 3.1. A von Neumann algebra on a Hilbert space $\mathcal{H}$ is a unital *-subalgebra $\mathcal{M}\subset \mathcal{B}(\mathcal{H})$ that is closed in the weak operator topology.

We will see (Theorem 4.1) that this is equivalent to closure in SOT, σ-weak, and several other topologies — but not the norm topology. A norm-closed unital *-subalgebra of $\mathcal{B}(\mathcal{H})$ is called a concrete C*-algebra; a vN algebra is “more closed” than that.

Why the definition uses WOT. Theorem 4.1 turns the choice of topology in Definition 3.1 into a convention: closure in WOT, closure in SOT, closure in the σ-weak topology and the algebraic condition $\mathcal{M}={\mathcal{M}}^{\prime \prime }$ single out the same unital *-subalgebras, so any of them could serve as the definition, and all of them are used in the literature. WOT is the natural one to write down. It is the coarsest of these topologies, so it has the most convergent nets, and being WOT-closed is the strongest of the closedness conditions: every SOT or σ-weak limit is also a WOT limit, and a WOT-closed algebra is therefore SOT-closed and σ-weakly closed with no theorem needed. The content of Theorem 4.1 runs in the opposite direction. A unital *-algebra that contains only the limits of a finer topology already contains every WOT limit.

WOT is also the topology in which closedness is easiest to check. Commutation is a statement about matrix elements, so a commutant is WOT-closed on sight (Lemma 3.3), and in AQFT the local algebras arrive as commutants and double commutants, such as the algebra $\{W(f):\mathrm{supp}f\subset \mathcal{O}{\}}^{\prime \prime }$ of Block C or the commutants that enter Haag duality. But the topology intrinsic to $\mathcal{M}$ is the σ-weak one. It is the weak-* topology defined by the predual (§6), and it is determined by $\mathcal{M}$ alone, while WOT depends on the Hilbert space on which $\mathcal{M}$ acts: under the isomorphism $x\mapsto x\otimes 1$ of $\mathcal{M}$ onto $\mathcal{M}\otimes 1\subset \mathcal{B}(\mathcal{H}\otimes {\ell }^2)$, the WOT of the larger space induces on $\mathcal{M}$ its σ-weak topology. This is why normal states are defined through σ-weak continuity (Definition 6.1).

Definition 3.2. The commutant of a subset $S\subset \mathcal{B}(\mathcal{H})$ is

$$S^{\prime }:=\{T\in \mathcal{B}(\mathcal{H}):Ta=aT\text{ for all }a\in S\}.$$

Physical picture. The commutant is the set of all operations compatible with $S$: everything that can be done to the system without disturbing any measurement of the observables in $S$. Two readings will recur. (1) Subsystem reading: if $S$ is the algebra of one subsystem, $S^{\prime }$ contains the operators of “everything else” — in the bipartite model $\mathcal{B}({\mathcal{H}}_1)\otimes 1$, the commutant is exactly $1\otimes \mathcal{B}({\mathcal{H}}_2)$ (Example 5.4). (2) Relativistic reading: in AQFT, microcausality says the algebra of a spacelike-separated region sits inside the commutant of the local algebra; Haag duality (Block C) is the statement that it exhausts the commutant — the causal complement is everything compatible. The commutant thus turns a kinematical notion (commutation) into a structural one (a second algebra), and the interplay $\mathcal{M}\leftrightarrow {\mathcal{M}}^{\prime }$ becomes, after Tomita–Takesaki (Week 5), a precise system–environment symmetry implemented by the modular conjugation $J$.

Lemma 3.3. [Proved.] The commutant $S^{\prime }$ of any subset is a unital algebra closed in WOT (hence a vN algebra when $S$ is self-adjoint, i.e., $S=S^{\ast }$).

Proof. Closure under sum, scalar multiplication, and product is by linearity of commutation. For closure under adjoint when $S^{\ast }=S$: if $T\in S^{\prime }$ and $a\in S$, then $a^{\ast }\in S$ (because $S$ is self-adjoint), so $T{a}^{\ast }=a^{\ast }T$. Taking adjoints, $a{T}^{\ast }=T^{\ast }a$ for all $a\in S$, i.e., $T^{\ast }\in S^{\prime }$. Closure in WOT: if $T_n\to T$ in WOT and $T_n\in S^{\prime }$, then for any $a\in S$, $⟨\eta ,(Ta-aT)\xi ⟩={\lim }_n⟨\eta ,(T_{n}a-a{T}_n)\xi ⟩=0,$ so $Ta=aT$, i.e., $T\in S^{\prime }$. $\square$

The double commutant $S^{\prime \prime }:=(S^{\prime }{)}^{\prime }$ is therefore also a vN algebra (when $S$ is self-adjoint). And $S\subset S^{\prime \prime }$ trivially (if $a\in S$ commutes with everything in $S^{\prime }$, then $a\in S^{\prime \prime }$), so $S\subset S^{\prime \prime }$ for every self-adjoint $S$.

4. The bicommutant theorem

Theorem 4.1 (von Neumann’s bicommutant theorem, 1929). [Proved — general case in §4.2; §4.1 works out the finite-dimensional structure. Refs: Bratteli–Robinson Vol. I §2.4.3.] Let $\mathcal{M}$ be a unital *-subalgebra of $\mathcal{B}(\mathcal{H})$. The following are equivalent:

(i) $\mathcal{M}={\mathcal{M}}^{\prime \prime }$; (ii) $\mathcal{M}$ is closed in WOT; (iii) $\mathcal{M}$ is closed in SOT; (iv) $\mathcal{M}$ is closed in the σ-weak topology.

This is the foundational theorem of von Neumann algebra theory: it identifies a purely algebraic condition (being equal to its double commutant) with a purely topological condition (being closed in WOT/SOT/σ-weak). It is what makes the theory tractable — to check that a concrete *-subalgebra is a vN algebra, we can use either characterization, whichever is easier in context.

Physical picture. The bicommutant theorem is a statement about what counts as an observable of a system once its basic observables are given. Start from a set $S$ of operators you declare measurable. The commutant $S^{\prime }$ is the set of all symmetries and external operations compatible with $S$. The theorem says: the operators obtainable from $S$ by taking limits of measurements (topological closure) are exactly the operators that respect every compatibility that $S$ respects (the double commutant). “What you can approximate by experiments on the system” = “what commutes with everything the system is independent of.” Neither side mentions the other’s structure — one is analysis, the other is algebra — and their identity is the non-trivial content. This duality is also why the construction $S\mapsto S^{\prime \prime }$ is the right notion of “the algebra generated by $S$”: it adds no operator that could be excluded on grounds of locality or compatibility.

4.1 Proof in the finite-dimensional case

The finite-dimensional case is the prototype and we work it out in full.

Setup. Let $\dim \mathcal{H}=n<\infty$. By Lemma 2.1, all topologies coincide; “closed” is automatic for any subspace. The non-trivial content of the theorem is then: every unital *-subalgebra of $M_n(\mathbb{C})$ equals its own double commutant.

Lemma 4.2 (Wedderburn). [Stated only — refs: any standard text on representation theory of finite-dimensional semisimple algebras (Maschke + Schur), e.g. Lang, Algebra, §XVII.] Let $\mathcal{M}\subset M_n(\mathbb{C})$ be a unital *-subalgebra. Then there is an orthogonal decomposition

$$\mathcal{H}=\underset{i=1}{\overset{k}{⨁}}{\mathcal{H}}_i,{\mathcal{H}}_i\cong {\mathbb{C}}^{m_i}\otimes {\mathbb{C}}^{k_i},$$

such that, with respect to this decomposition,

$$\mathcal{M}\cong \underset{i=1}{\overset{k}{⨁}}{M}_{m_i}(\mathbb{C})\otimes 1_{k_i}$$

— $\mathcal{M}$ acts as $a_i\otimes 1$ on ${\mathcal{H}}_i$ for some $a_i\in M_{m_i}(\mathbb{C})$, and the $a_i$ are arbitrary.

We do not prove Wedderburn here; it follows from the basic representation theory of finite-dimensional semisimple algebras over $\mathbb{C}$ (Maschke + Schur). The key structural takeaway: any unital *-subalgebra of $M_n(\mathbb{C})$ is determined by a list of pairs $\{(m_i,k_i)\}$ with $\sum m_i{k}_i=n$.

Computing the commutant. Given the Wedderburn decomposition, we compute ${\mathcal{M}}^{\prime }$:

$$T\in {\mathcal{M}}^{\prime }⟺T\text{ commutes with all }a\otimes 1\text{ on each }{\mathcal{H}}_i.$$

On ${\mathcal{H}}_i={\mathbb{C}}^{m_i}\otimes {\mathbb{C}}^{k_i}$, an operator commuting with all $M_{m_i}(\mathbb{C})\otimes 1$ must (by Schur’s lemma in tensor form) be of the form $1\otimes b_i$ for some $b_i\in M_{k_i}(\mathbb{C})$. Hence

$${\mathcal{M}}^{\prime }\cong \underset{i=1}{\overset{k}{⨁}}{1}_{m_i}\otimes M_{k_i}(\mathbb{C}).$$

And by symmetry, applying the same calculation to ${\mathcal{M}}^{\prime }$,

$${\mathcal{M}}^{\prime \prime }\cong \underset{i=1}{\overset{k}{⨁}}{M}_{m_i}(\mathbb{C})\otimes 1_{k_i}=\mathcal{M}.$$

This proves the bicommutant theorem in finite dimensions. $\square$

The proof is satisfying because it makes vN algebra theory in finite dimensions completely explicit: a vN subalgebra of $M_n(\mathbb{C})$ is just a direct-sum-of-matrix-algebras structure determined by the dimensions $\{m_i,k_i\}$. The commutant comes for free by switching the roles of “intrinsic” and “multiplicity” indices.

Worked example: the diagonal algebra in $M_3(\mathbb{C})$

To make the bicommutant theorem concrete, take $\mathcal{M}\subset M_3(\mathbb{C})$ to be the algebra of diagonal matrices:

$$\mathcal{M}=\{\mathrm{diag}({\alpha }_1,{\alpha }_2,{\alpha }_3):{\alpha }_i\in \mathbb{C}\}\cong \mathbb{C}\oplus \mathbb{C}\oplus \mathbb{C}.$$

This is the Wedderburn decomposition ${⨁}_{i=1}^3{M}_1(\mathbb{C})\otimes 1_1$ (three blocks, each $1\times 1$, with no multiplicity).

Computing ${\mathcal{M}}^{\prime }$. A matrix $T=(T_{ij})\in M_3(\mathbb{C})$ commutes with every diagonal $\mathrm{diag}({\alpha }_1,{\alpha }_2,{\alpha }_3)$ iff $T_{ij}{\alpha }_j={\alpha }_i{T}_{ij}$ for all $i,j$ and all choices of ${\alpha }_k$. For $i\ne j$, choosing ${\alpha }_j=1,{\alpha }_i=0$ forces $T_{ij}=0$. So ${\mathcal{M}}^{\prime }=\mathcal{M}$ — the diagonal algebra is its own commutant. (This is the prototype “maximal abelian subalgebra,” to be developed in §7 for arbitrary spectra.)

Computing ${\mathcal{M}}^{\prime \prime }$. Since ${\mathcal{M}}^{\prime }=\mathcal{M}$, the double commutant is ${\mathcal{M}}^{\prime \prime }=({\mathcal{M}}^{\prime }{)}^{\prime }={\mathcal{M}}^{\prime }=\mathcal{M}$. So $\mathcal{M}={\mathcal{M}}^{\prime \prime }$, confirming the bicommutant theorem in this case.

Comparison: a non-trivial multiplicity. Now take $\mathcal{N}\subset M_3(\mathbb{C})$ given by

$$\mathcal{N}=\{\mathrm{diag}(\alpha ,\alpha ,\beta ):\alpha ,\beta \in \mathbb{C}\}\cong \mathbb{C}\oplus \mathbb{C}.$$

This is the Wedderburn decomposition $M_1(\mathbb{C})\otimes 1_2\oplus M_1(\mathbb{C})\otimes 1_1$ — two blocks, the first with multiplicity 2, the second with multiplicity 1.

A computation similar to above gives

$${\mathcal{N}}^{\prime }=\{\left(\begin{array}{cc}A& 0\\ 0& \gamma \end{array}\right):A\in M_2(\mathbb{C}),\gamma \in \mathbb{C}\}\cong M_2(\mathbb{C})\oplus \mathbb{C}.$$

The multiplicity-2 block in $\mathcal{N}$ allows ${\mathcal{N}}^{\prime }$ to “see” the full $M_2(\mathbb{C})$ acting on the corresponding two-dimensional subspace. The multiplicity-1 block leaves only $\mathbb{C}$ in the commutant.

Computing ${\mathcal{N}}^{\prime \prime }$: take the commutant of $M_2(\mathbb{C})\oplus \mathbb{C}$ inside $M_3$. This is $\mathbb{C}\cdot 1_2\oplus \mathbb{C}=\mathcal{N}$. So again ${\mathcal{N}}^{\prime \prime }=\mathcal{N}$, and the bicommutant theorem is verified.

The contrast between $\mathcal{M}$ (no multiplicity) and $\mathcal{N}$ (multiplicity 2 in the first block) shows the role of the $k_i$ indices in Wedderburn: they are exactly the dimensions of the commutant blocks. Algebra and commutant are dual, with intrinsic and multiplicity dimensions swapping roles.

4.2 Proof of the general case

The general proof is short and uses no structure theory. It is the standard argument going back to von Neumann, it rests on one observation about projections, and it proves the four equivalences at once by comparing closures. In finite dimensions it gives a second proof of §4.1 that avoids Wedderburn; what §4.1 adds is the explicit form of $\mathcal{M}$ and ${\mathcal{M}}^{\prime }$.

For a subset $X\subset \mathcal{B}(\mathcal{H})$ we write ${\overline{X}}^{\tau }$ for its closure in the topology $\tau$, and σs for the σ-strong topology, defined by the seminorms $T\mapsto ({\sum }_i\Vert T{\xi }_i{\Vert }^2{)}^{1/2}$ with ${\sum }_i\Vert {\xi }_i{\Vert }^2<\infty$. Taking finitely many nonzero ${\xi }_i$ gives back the SOT seminorms, and the Cauchy–Schwarz inequality bounds every σ-weak seminorm by a σ-strong one, so σs is finer than both SOT and σ-weak.

Step 1 (the chain of closures). Let ${\mathcal{M}}_0$ be a unital *-subalgebra of $\mathcal{B}(\mathcal{H})$. A finer topology has fewer convergent nets and therefore smaller closures, and ${\mathcal{M}}_0^{\prime \prime }$ is WOT-closed (Lemma 3.3) and contains ${\mathcal{M}}_0$. Thus

$${\mathcal{M}}_0\subset {\overline{{\mathcal{M}}_0}}^{\sigma s}\subset {\overline{{\mathcal{M}}_0}}^{\mathrm{SOT}}\subset {\overline{{\mathcal{M}}_0}}^{\mathrm{WOT}}\subset {\mathcal{M}}_0^{\prime \prime },$$

and the same chain holds with the σ-weak closure in place of the SOT closure. Every inclusion here is elementary. The theorem follows once we close the chain, that is, once we show ${\mathcal{M}}_0^{\prime \prime }\subset {\overline{{\mathcal{M}}_0}}^{\sigma s}$.

Step 2 (one vector). Take $x\in {\mathcal{M}}_0^{\prime \prime }$ and $\xi \in \mathcal{H}$, and let $P$ be the orthogonal projection onto the closed subspace $\mathcal{K}:=\overline{{\mathcal{M}}_0\xi }$. The subspace $\mathcal{K}$ is invariant under ${\mathcal{M}}_0$, and so is ${\mathcal{K}}^{\perp }$: for $\eta \in {\mathcal{K}}^{\perp }$, $a\in {\mathcal{M}}_0$ and $k\in \mathcal{K}$ we have $⟨a\eta ,k⟩=⟨\eta ,a^{\ast }k⟩=0$, because $a^{\ast }\in {\mathcal{M}}_0$. Therefore $P\in {\mathcal{M}}_0^{\prime }$, and $x$ commutes with $P$. Since $1\in {\mathcal{M}}_0$, the vector $\xi$ lies in $\mathcal{K}$, and

$$x\xi =xP\xi =Px\xi \in \mathcal{K}.$$

That is, $x\xi$ is a norm limit of vectors $a\xi$ with $a\in {\mathcal{M}}_0$.

Step 3 (amplification). To control infinitely many vectors at once, consider $\tilde{\mathcal{H}}:={⨁}_{i\ge 1}\mathcal{H}\cong \mathcal{H}\otimes {\ell }^2$, on which $a\in {\mathcal{M}}_0$ acts diagonally as $a\otimes 1$. A bounded operator on $\tilde{\mathcal{H}}$ is a matrix $(T_{ij})$ of operators on $\mathcal{H}$, and it commutes with every $a\otimes 1$ exactly when each entry $T_{ij}$ lies in ${\mathcal{M}}_0^{\prime }$. The operator $x\otimes 1$ commutes with all such matrices, so $x\otimes 1\in ({\mathcal{M}}_0\otimes 1{)}^{\prime \prime }$. Applying Step 2 to the unital *-algebra ${\mathcal{M}}_0\otimes 1$ and to the vector $\tilde{\xi }=({\xi }_1,{\xi }_2,\dots )$, with ${\sum }_i\Vert {\xi }_i{\Vert }^2<\infty$, gives for every $ϵ>0$ an element $a\in {\mathcal{M}}_0$ with

$$\sum _i\Vert (x-a){\xi }_i{\Vert }^2<{ϵ}^2,$$

which is precisely the statement that $x$ lies in the σ-strong closure of ${\mathcal{M}}_0$.

The chain of Step 1 now closes up, and all its members coincide:

$${\overline{{\mathcal{M}}_0}}^{\sigma s}={\overline{{\mathcal{M}}_0}}^{\mathrm{SOT}}={\overline{{\mathcal{M}}_0}}^{\sigma \text{-weak}}={\overline{{\mathcal{M}}_0}}^{\mathrm{WOT}}={\mathcal{M}}_0^{\prime \prime }.$$

This is a sharper form of Theorem 4.1: for a unital *-algebra, the closure in any of these topologies is the bicommutant. The four conditions of the theorem follow at once, since $\mathcal{M}$ is closed in one of these topologies exactly when it equals its closure there, that is, exactly when $\mathcal{M}={\mathcal{M}}^{\prime \prime }$. $\square$

Note where the hypotheses entered: the *-operation made $P$ commute with ${\mathcal{M}}_0$, and the unit placed $\xi$ in $\mathcal{K}$. The norm topology is absent from the chain. The norm closure of ${\mathcal{M}}_0$ is its C*-closure, in general strictly smaller than ${\mathcal{M}}_0^{\prime \prime }$, and that gap is what §1 set out to fill.

The equalities in the chain have different origins. For every convex set the WOT- and SOT-closures agree: both topologies have the same continuous linear functionals, the finite sums $T\mapsto {\sum }_{k=1}^n⟨{\eta }_k,T{\xi }_k⟩$, and by the Hahn–Banach theorem a closed convex set is determined by its continuous linear functionals. The comparison with the σ-weak topology is of a different kind. For a trace-class $\rho$ of infinite rank, the kernel of $T\mapsto \mathrm{Tr}(\rho T)$ is a σ-weakly closed subspace that is WOT-dense in $\mathcal{B}(\mathcal{H})$ (Problem 8*). For *-algebras the equality of the σ-weak and WOT closures is therefore a genuine consequence of the algebraic structure, and it enters through Steps 2 and 3.

Once ${\mathcal{M}}_0^{\prime \prime }$ is known to be the strong closure of ${\mathcal{M}}_0$, we can ask for approximants of controlled norm. This is the content of Kaplansky’s density theorem, which refines Theorem 4.1 and comes after it, both historically and logically:

Theorem 4.3 (Kaplansky 1951). [Stated only — refs: Kaplansky, “A theorem on rings of operators,” Pacific J. Math. 1 (1951) 227–232, Theorem 1; Bratteli–Robinson Vol. I §2.4.3.] For any unital *-subalgebra ${\mathcal{M}}_0\subset \mathcal{B}(\mathcal{H})$, the unit ball $\{a\in {\mathcal{M}}_0^{\prime \prime }:\Vert a\Vert \le 1\}$ is the SOT-closure of the unit ball $\{a\in {\mathcal{M}}_0:\Vert a\Vert \le 1\}$.

Step 2 gives approximants with no control on their norms, and Kaplansky’s theorem supplies approximants with $\Vert a_{\alpha }\Vert \le \Vert x\Vert$. This matters in practice. On norm-bounded sets WOT and σ-weak coincide, and so do SOT and σ-strong, so bounded approximation makes the differences between these topologies harmless; and when $\mathcal{H}$ is separable the unit ball is metrizable in SOT, so the approximating net can be replaced by a sequence.

The proof starts from Theorem 4.1. For a self-adjoint $x\in {\mathcal{M}}_0^{\prime \prime }$ with $\Vert x\Vert \le 1$, we first obtain self-adjoint approximants: if $a_{\alpha }\to x$ strongly, then $\frac{1}{2}(a_{\alpha }+a_{\alpha }^{\ast })\to x$ weakly, and since the weak and strong closures of the convex set of self-adjoint elements of ${\mathcal{M}}_0$ agree, these elements are strongly dense in the self-adjoint part of ${\mathcal{M}}_0^{\prime \prime }$. Next we write $x=f(c)$ with

$$f(t)=\frac{2t}{1+t^2},$$

where $c=c^{\ast }\in {\mathcal{M}}_0^{\prime \prime }$ exists because $f$ maps $[-1,1]$ increasingly onto itself. We approximate $c$ strongly by self-adjoint $c_{\alpha }\in {\mathcal{M}}_0$ and use that $c\mapsto f(c)$ is strongly continuous on self-adjoint operators. The elements $f(c_{\alpha })$ lie in the C*-closure of ${\mathcal{M}}_0$, have norm at most one because $|f|\le 1$ on $\mathbb{R}$, and converge strongly to $x$; a norm approximation inside the unit ball then moves them into ${\mathcal{M}}_0$ itself. The general case follows by applying this to the self-adjoint operator

$$\left(\begin{array}{cc}0& x\\ x^{\ast }& 0\end{array}\right)$$

acting on $\mathcal{H}\oplus \mathcal{H}$, and reading off the off-diagonal entry of its approximants.

The cost: we need the C*-closure of ${\mathcal{M}}_0$ to have the continuous functional calculus, which requires the C*-identity. This is one place where the C*-structure (Week 1) is genuinely used to do the vN algebra theory of Week 2.

4.3 Why bicommutant matters

The practical consequence of Theorem 4.1 is that we have two complementary ways to describe vN algebras:

• Topologically: as WOT-closed unital *-subalgebras.
• Algebraically: as $S^{\prime \prime }$ for some self-adjoint subset $S$.

The second description is constructive: given any “interesting” set of operators, the vN algebra they generate is just $S^{\prime \prime }$. For example, the vN algebra $W^{\ast }(a)$ generated by a single self-adjoint operator $a$ is $\{a{\}}^{\prime \prime }$, and we will compute it explicitly in §7 below.

5. Examples and key constructions

Example 5.1 ($\mathcal{B}(\mathcal{H})$). $\mathcal{B}(\mathcal{H})$ is itself a vN algebra (trivially: it is its own SOT-closure). Its commutant is the center of $\mathcal{B}(\mathcal{H})$, which is $\mathbb{C}\cdot 1$: the only operators that commute with all bounded operators are scalar multiples of the identity. Hence $\mathcal{B}(\mathcal{H})$ is a factor:

Definition 5.2. A vN algebra $\mathcal{M}$ is a factor if its center $\mathcal{Z}(\mathcal{M}):=\mathcal{M}\cap {\mathcal{M}}^{\prime }$ is trivial: $\mathcal{Z}(\mathcal{M})=\mathbb{C}\cdot 1$.

Factors are the “irreducible building blocks” of vN algebra theory: the type classification (Week 3) is a classification of factors. Under the standard measurable-field hypotheses used throughout this course—in particular, for von Neumann algebras with separable predual in their standard representation—one has a central decomposition as a direct integral of factors over the spectrum of the center. Without those countability/standardness hypotheses, the slogan still points in the right direction, but its precise formulation requires the general central-decomposition formalism rather than an elementary direct integral. We use the separable-predual version without proof.

Physical picture. The center is the classical part of the theory: central elements commute with every observable, so they can be measured without disturbing anything — they behave like classical parameters, not quantum observables. Nontrivial center means superselection sectors: the value of a central observable (total electric charge, a θ-angle, the phase of a system across a phase transition) is fixed within each sector, and no physical operation creates superpositions across sectors. A factor is a theory with no such classical labels — fully quantum, no superselection. The direct-integral decomposition over the center is then the precise statement that any quantum system is a classical mixture of fully quantum ones: first diagonalize the classical data, then study each fixed-classical-value world separately. This is why classifying factors classifies everything.

Example 5.3 ($L^{\infty }(X,\mu )$ as a maximal abelian vN algebra). Let $(X,\mu )$ be a $\sigma$-finite measure space, and let $\mathcal{H}=L^2(X,\mu )$. Then $L^{\infty }(X,\mu )$ acts on $\mathcal{H}$ by multiplication: $(M_f\xi )(x)=f(x)\xi (x)$.

Claim. $L^{\infty }(X,\mu {)}^{\prime }=L^{\infty }(X,\mu )$ (this algebra is its own commutant).

Proof of claim. Suppose $T\in L^{\infty }(X,\mu {)}^{\prime }$. We show $T=M_g$ for some $g\in L^{\infty }$.

Pick a measurable $E\subset X$ with $\mu (E)<\infty$ (uses $\sigma$-finiteness) and set ${\xi }_E=1_E\in L^2$. Define $g_E:=T{\xi }_E$, an element of $L^2$. For any $f\in L^{\infty }$ with $\mathrm{supp}(f)\subset E$,

$$T(f{\xi }_E)=T(M_f{\xi }_E)=M_{f}T({\xi }_E)=M_f{g}_E=f{g}_E.$$

But also, $f{\xi }_E=f\cdot 1_E=f$ (since $\mathrm{supp}(f)\subset E$). So $Tf=f{g}_E$ for every such $f$.

By exhausting $X$ with $\sigma$-finite increasing sets $\{E_n\}$, the functions $g_{E_n}$ are consistent, defining a measurable $g$ on $X$ with $Tf=M_{g}f$ for all $f\in L^2$ with bounded support. This extends by density to $L^2$. The boundedness of $T$ as an operator on $L^2$ implies $g\in L^{\infty }$ with $\Vert g{\Vert }_{\infty }=\Vert T\Vert$. Hence $T=M_g\in L^{\infty }(X,\mu )$. $\square$

The maximality $L^{\infty }(X,\mu {)}^{\prime }=L^{\infty }(X,\mu )$ is what makes commutative vN algebras tractable: they coincide with their commutants, hence their double commutants — they are vN algebras automatically. This is the deep connection between measure theory and vN algebra theory: every commutative vN algebra is $L^{\infty }$ of some measure space (Gelfand–Naimark for vN), and conversely.

Physical picture. This example fixes the dictionary that runs the whole subject: commutative vN algebra = classical probability theory in operator clothing. $L^{\infty }(X,\mu )$ is the algebra of bounded random variables; normal states on it are probability densities ($L^1$ functions, Example 6.4); spectral projections are events; the lattice of projections is the σ-algebra. Von Neumann algebra theory is then precisely noncommutative measure theory — the slogan is exact, not loose: drop commutativity from $L^{\infty }$ and you get vN algebras; drop it from $C(X)$ and you get C*-algebras (noncommutative topology). Week 3’s type classification answers the question this dictionary immediately raises: what kinds of “noncommutative measure spaces” exist that have no classical counterpart? Types II and III are the answer.

Example 5.4 (Tensor product commutant theorem). Let $\mathcal{H}={\mathcal{H}}_1\otimes {\mathcal{H}}_2$, and let ${\mathcal{M}}_i\subset \mathcal{B}({\mathcal{H}}_i)$ be vN algebras. Then

$$({\mathcal{M}}_1\otimes {\mathcal{M}}_2{)}^{\prime }={\mathcal{M}}_1^{\prime }\otimes {\mathcal{M}}_2^{\prime },$$

where the tensor product on the left is the WOT-closure of the algebraic tensor product. This is Tomita’s commutation theorem.

In the special case ${\mathcal{M}}_1=\mathcal{B}({\mathcal{H}}_1),{\mathcal{M}}_2=1$:

$$(\mathcal{B}({\mathcal{H}}_1)\otimes 1{)}^{\prime }=1\otimes \mathcal{B}({\mathcal{H}}_2).$$

Proof in finite dimensions. Take ${\mathcal{H}}_i={\mathbb{C}}^{n_i}$. By Wedderburn (Lemma 4.2) applied to ${\mathcal{M}}_1\otimes 1$, the commutant is ${\mathcal{M}}_1^{\prime }\otimes \mathcal{B}({\mathcal{H}}_2)$. Iterating: $({\mathcal{M}}_1\otimes {\mathcal{M}}_2{)}^{\prime }={\mathcal{M}}_1^{\prime }\otimes {\mathcal{M}}_2^{\prime }$ by the same argument. $\square$

Warning — what this theorem is not in AQFT. The tensor commutant theorem above presumes a spatial tensor-product structure $\mathcal{H}={\mathcal{H}}_1\otimes {\mathcal{H}}_2$. This is the right model for finite-dimensional bipartite quantum systems and for type-I situations where the Hilbert space genuinely factors. It is not the right model for AQFT in type III.

In a relativistic QFT, when a Cauchy slice $\Sigma$ is decomposed into spacelike-disjoint regions ${\Sigma }_1,{\Sigma }_2$, the corresponding local algebras $\mathcal{A}({\Sigma }_1),\mathcal{A}({\Sigma }_2)$ satisfy commutation (microcausality / locality) and often Haag duality ($\mathcal{A}({\Sigma }_1{)}^{\prime }=\mathcal{A}({\Sigma }_2)$ for an appropriate causal complement) — but the underlying Hilbert space generally does not factor as ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$. The local algebras are type III${}_1$ (Block C); type III algebras admit no minimal projections, so the identification with operators-on-a-tensor-factor fails in a strong structural sense.

A tensor factorization of the algebra structure does emerge under the additional split property (Doplicher–Longo, Buchholz, Wichmann) and nuclearity assumptions, applied to algebras of well-separated regions. But this is a theorem requiring real hypotheses, not a corollary of the spatial commutant theorem above.

So: keep this theorem as a finite-dimensional / type-I model. In AQFT use locality, Haag duality, and (where applicable) the split property — not automatic tensor factorization.

Example 5.5 (Group von Neumann algebras). Let $G$ be a discrete group, and let $\lambda :G\to \mathcal{B}({\ell }^2(G))$ be the left-regular representation: $(\lambda (g)\xi )(h)=\xi (g^{-1}h)$. The vN algebra

$$\mathcal{L}(G):=\{\lambda (g):g\in G{\}}^{\prime \prime }\subset \mathcal{B}({\ell }^2(G))$$

is the group vN algebra of $G$. It is a fundamental construction with deep ties to representation theory.

Properties:

1. $\mathcal{L}(G)$ admits a tracial vector state. The trace is $\tau (a)=⟨{\delta }_e,a{\delta }_e⟩$ where ${\delta }_e\in {\ell }^2(G)$ is the indicator of the identity element. Tracial: for any $g,h$, $\tau (\lambda (g)\lambda (h))=\tau (\lambda (gh))=[gh=e]$, while $\tau (\lambda (h)\lambda (g))=[hg=e]$, and $gh=e⟺hg=e$. So $\tau$ is tracial on the linear span of $\{\lambda (g)\}$, hence on $\mathcal{L}(G)$.
2. $\mathcal{L}(G)$ is a factor iff $G$ is icc (every non-identity conjugacy class is infinite). Examples of icc: $S_{\infty }$ (finitary permutations), free groups $F_n$ for $n\ge 2$, ${\mathrm{PSL}}_2(\mathbb{Z})$.
3. For abelian $G$, $\mathcal{L}(G)\cong L^{\infty }(\hat{G})$ (Pontryagin dual), via Fourier transform. So abelian group vN algebras are commutative — never factors (unless $G$ is trivial).
4. For icc $G$, $\mathcal{L}(G)$ is a type II${}_1$ factor (proved in Week 3). This is one of the most flexible sources of type II${}_1$ factors.

Group vN algebras are a backbone of the theory of ${\mathrm{II}}_1$ factors and of later rigidity theory. They also give us a concrete counterweight to the hyperfinite example of Week 3; they should not be conflated with the separate modular theory used to classify type III factors.

Example 5.6 (Spectral projections live in $\mathcal{M}$). If $a\in \mathcal{M}$ is self-adjoint, every spectral projection $1_E(a)$ for Borel $E\subset \sigma (a)$ is in $\mathcal{M}$.

Reason. The bounded Borel functional calculus gives $1_E(a)=E_a(E)$. To see it as a strong limit, first approximate indicators of intervals by uniformly bounded continuous functions converging pointwise, and apply dominated convergence to each scalar spectral measure ${\mu }_{\xi }(B)=⟨\xi ,E_a(B)\xi ⟩$:

$$\Vert (f_n(a)-1_E(a))\xi {\Vert }^2={\int }_{\sigma (a)}|f_n-1_E{|}^{2}d{\mu }_{\xi }⟶0.$$

A monotone-class argument extends the conclusion to every Borel $E$. Each $f_n(a)$ belongs to the norm-closed C*-algebra generated by $a$, hence to $\mathcal{M}$; SOT-closedness then gives $1_E(a)\in \mathcal{M}$. $\square$

This is the crucial property the C*-closure does not have, and the most immediately useful consequence of Theorem 4.1.

6. The predual

Definition 6.1. A linear functional $\omega :\mathcal{M}\to \mathbb{C}$ is normal if it is continuous in the σ-weak topology (equivalently: continuous on bounded sets in WOT).

A non-normal state lies outside the folium selected by the given representation: on $\mathcal{B}(\mathcal{H})$ it is not described by a trace-class density operator. Some examples are constructed by ultrafilters, while others can arise as singular infinite-volume or limiting states. They are mathematically real and should not simply be called impossible; normal states are the standing class here because they respect the weak limits used in the represented theory.

Physical picture. Normality is the noncommutative version of countable additivity in probability theory. A normal state respects monotone limits: if an increasing net of yes/no questions (projections) has supremum $p$, the probability assigned to $p$ is the supremum of the probabilities — exactly as a σ-additive measure treats an increasing union of events. A non-normal state can be internally consistent while being discontinuous for this monotone structure. In Week 4 and beyond, “faithful normal state” is the standing hypothesis of the vector-state form of modular theory: normality ties the state to the weak topology and predual of the represented algebra.

Theorem 6.2 (Sakai). [Stated only — refs: Sakai, C*-algebras and W*-algebras, ch. 1.] Every vN algebra $\mathcal{M}$ has a unique Banach space of normal linear functionals, called the predual ${\mathcal{M}}_{\ast }$, such that $\mathcal{M}=({\mathcal{M}}_{\ast }{)}^{\ast }$ as Banach spaces.

This is the abstract characterization of vN algebras: a C*-algebra is *-isomorphic to a vN algebra iff it is the dual of some Banach space. (Sakai’s theorem, 1956; such C*-algebras are called W*-algebras.) It tells us that vN algebras are intrinsically defined — we don’t need to fix a specific Hilbert-space realization to know that $\mathcal{M}$ is “abstractly” a vN algebra.

Example 6.3. $\mathcal{B}(\mathcal{H}{)}_{\ast }=\mathcal{T}(\mathcal{H})$ (trace-class operators), and the duality is ${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a)$ for $\rho$ trace-class. Normal states on $\mathcal{B}(\mathcal{H})$ are exactly density-matrix states. Non-normal states do exist (as functionals on $\mathcal{B}(\mathcal{H})$) but they are not represented by any $\rho \in \mathcal{T}(\mathcal{H})$.

Example 6.4. $L^{\infty }(X,\mu {)}_{\ast }=L^1(X,\mu )$, with the duality ${\omega }_f(g)=\int fgd\mu$. Normal states are absolutely continuous probability measures.

Example 6.5. For the hyperfinite II${}_1$ factor $\mathcal{R}$ (Week 3), the predual ${\mathcal{R}}_{\ast }$ is more subtle — it consists of functionals $\omega (a)=\tau (\rho a)$ for “noncommutative $L^1$” densities $\rho$. The trace $\tau$ itself is a normal state (in fact the trace), and ${\mathcal{R}}_{\ast }$ is built around it.

The predual is the natural “state space” of a vN algebra: it contains the physically meaningful states and allows one to talk about convergence of states in a reasonable sense.

7. Worked computation: the vN algebra of a self-adjoint operator

Let $a=a^{\ast }\in \mathcal{B}(\mathcal{H})$. By the spectral theorem, there is a projection-valued measure $E_a:\mathcal{B}(\sigma (a))\to \mathcal{B}(\mathcal{H})$ such that $a={\int }_{\sigma (a)}\lambda d{E}_a(\lambda )$.

Theorem 7.1. [Sketched — via spectral theorem and bounded Borel functional calculus.] The vN algebra generated by $a$ is

$$W^{\ast }(a):=\{a{\}}^{\prime \prime }=\{f(a):f\in L^{\infty }(\sigma (a),E_a)\}$$

— the algebra of bounded measurable functions of $a$, viewed as multiplication operators via the spectral theorem.

Proof sketch. The argument has two layers — a continuous functional calculus from C*-theory, then a bounded Borel functional calculus from the spectral theorem.

Step 1 (continuous functional calculus). Polynomials in $a$ are in $W^{\ast }(a)$ by definition. By Stone–Weierstrass, polynomials are uniformly (norm-) dense in $C(\sigma (a))$, so the continuous functional calculus gives $\{f(a):f\in C(\sigma (a))\}\subset W^{\ast }(a)$.

Step 2 (extending to bounded Borel functions via the spectral theorem). The spectral theorem for self-adjoint $a$ produces a projection-valued measure $E_a$ on Borel subsets of $\sigma (a)$, and a bounded Borel functional calculus $f\mapsto f(a):=\int fd{E}_a$ defined for every bounded Borel $f\in L^{\infty }(\sigma (a),E_a)$. The output $f(a)$ is in the double commutant of $a$: for any $T\in \{a{\}}^{\prime }$, one has $T{E}_a(B)=E_a(B)T$ for every Borel $B$ (a standard consequence of the spectral theorem), hence $Tf(a)=f(a)T$ for every bounded Borel $f$. So $f(a)\in \{a{\}}^{\prime \prime }=W^{\ast }(a)$.

This gives $\{f(a):f\in L^{\infty }(\sigma (a),E_a)\}\subset W^{\ast }(a)$. Note: the inclusion does not go through “polynomials SOT-approximate bounded Borel functions”; that statement is true but follows from the spectral theorem rather than from Stone–Weierstrass directly.

Reverse inclusion. Every element of $W^{\ast }(a)$ commutes with $\{a{\}}^{\prime }$. By the same spectral-theorem reasoning, every operator commuting with $\{a{\}}^{\prime }$ on $\mathcal{H}$ is a bounded Borel function of $a$. Hence $W^{\ast }(a)\subset \{f(a):f\in L^{\infty }(\sigma (a),E_a)\}$. $\square$

This is a structural fact: the vN algebra of a single self-adjoint operator is commutative, and is $L^{\infty }$ of the spectrum. The non-commutativity of vN algebras enters only when we have several non-commuting operators.

Corollary 7.2. [Stated only — follows from spectral multiplicity theory; refs: Bratteli–Robinson Vol. I §2.4.4.] $W^{\ast }(a)$ is a maximal abelian vN algebra iff the spectrum of $a$ is multiplicity-free (i.e., the spectral measure $E_a$ has no non-trivial “multiplicity”).

Example 7.3. Let $a$ be multiplication by $x$ on $L^2[0,1]$. Its spectrum is $[0,1]$ and is multiplicity-free, so $W^{\ast }(a)=L^{\infty }[0,1]$ acting by multiplication. This is the canonical model of a maximal abelian vN algebra.

Example 7.4 (Multiplicity issue). For $a=\mathrm{diag}(0,0,1)\in M_3(\mathbb{C})$: $\sigma (a)=\{0,1\}$, with multiplicities 2 and 1. $W^{\ast }(a)$ consists of operators of the form $\mathrm{diag}(\alpha ,\alpha ,\beta )$ — only 2-dimensional — which is not maximal abelian in $M_3(\mathbb{C})$. The maximal abelian extension is $\mathrm{diag}(\alpha ,\beta ,\gamma )$, which is 3-dimensional and corresponds to a “multiplicity-resolving” version of $a$.

8. Generation by sets of operators

Definition 8.1. Given a self-adjoint subset $S\subset \mathcal{B}(\mathcal{H})$, the vN algebra generated by $S$ is $S^{\prime \prime }$.

Lemma 8.2. [Sketched — direct from Theorem 4.1.] $S^{\prime \prime }$ equals the WOT-closure of the unital *-algebra generated by $S$.

(Reason: $S^{\prime \prime }$ is WOT-closed and contains the algebra generated by $S$. Conversely, the WOT-closure of any unital *-subalgebra equals its double commutant, by the sharper form of Theorem 4.1 proved in §4.2.)

Example 8.3 (Two non-commuting projections). Let $p,q\in \mathcal{B}(\mathcal{H})$ be two projections. The algebra $\{p,q{\}}^{\prime \prime }$ is non-commutative in general — and its structure is surprisingly rich. In finite dimensions ($\mathcal{H}={\mathbb{C}}^n$), this leads to the Halmos two-projection theorem: $\{p,q{\}}^{\prime \prime }$ decomposes the Hilbert space into common 1-dimensional pieces, on which each projection acts by $0$ or $1$, together with 2-dimensional blocks on which $p$ and $q$ are rank-one orthogonal projections at a definite angle. The 2-dimensional blocks carry the continuous parameter measuring that angle.

This is a useful finite-dimensional laboratory for the difference between the algebra generated by observables and the commutant that records their multiplicities.

9. What to take away

1. vN algebras are C*-algebras closed in a weak topology. The definition is concrete, but by Sakai’s theorem a vN algebra is intrinsically a C*-algebra that is a dual Banach space, and its σ-weak topology and normal states do not depend on the representation. Closing in WOT/SOT instead of norm gives access to spectral projections, and via the bicommutant theorem, to the full algebraic-topological duality.

2. The bicommutant theorem. Algebraic ($\mathcal{M}={\mathcal{M}}^{\prime \prime }$) ↔ topological (WOT/SOT/σ-weak-closed). This is the central structural result: it allows us to construct vN algebras as $S^{\prime \prime }$ for any self-adjoint $S$, and to verify the vN-algebra property either algebraically or topologically. For a unital *-algebra the WOT-, SOT- and σ-weak closures all equal the bicommutant, although the three topologies remain different, and this is why the choice of WOT in the definition is a convention.

3. Factors are the building blocks. With separable predual and the standard measurable-field setup, a vN algebra decomposes as a direct integral of factors over its center; in full generality one uses the corresponding central-decomposition formalism. The type classification (next week) classifies the factor fibers.

4. The predual encodes normal states. The predual ${\mathcal{M}}_{\ast }$ is canonically defined. Normal states are the states compatible with monotone limits of projections and are the standard class used for local QFT and statistical mechanics. Singular states can still be mathematically meaningful, but they lie outside this normal folium and must not be represented automatically by density operators.

5. Spectral projections are in $\mathcal{M}$. This is the property that makes vN algebras useful for measurement theory and entropy calculations.

6. The vN algebra of a single operator is commutative, and equals $L^{\infty }$ of its spectrum. Non-commutativity enters only with several non-commuting generators.

10. Looking ahead: Week 3

Next week we classify factors into types I, II, and III using the order structure on projections. Type I is the “ordinary” case — every type I factor is $\mathcal{B}(\mathcal{K})$ for some Hilbert space $\mathcal{K}$. Type II admits a trace and houses the hyperfinite II${}_1$ factor and group vN algebras of icc groups. Type III admits no nonzero normal semifinite trace and is the type obtained for standard local-QFT examples under the additional hypotheses made explicit in Week 12.

11. Problem set

Core problems.

1. Operator topologies. (a) Show that for any unital *-subalgebra ${\mathcal{M}}_0\subset \mathcal{B}(\mathcal{H})$, the WOT-closure and the SOT-closure of ${\mathcal{M}}_0$ coincide (and both equal ${\mathcal{M}}_0^{\prime \prime }$). (Hint: for the coincidence, show that WOT and SOT have the same continuous linear functionals and apply the Hahn–Banach theorem, which only needs ${\mathcal{M}}_0$ to be convex; for the identification with ${\mathcal{M}}_0^{\prime \prime }$, follow §4.2.) Note: the analogous statement for arbitrary bounded sets is FALSE — exhibit a counterexample using $\{S^n:n\ge 0\}$ for the unilateral shift (this set is bounded, $0$ is in its WOT-closure since $S^n\to 0$ in WOT, but $\Vert S^n\xi \Vert =\Vert \xi \Vert$ shows $0$ is not in its SOT-closure). What the first part uses is convexity, which $\{S^n\}$ lacks; the *-algebra structure is what identifies the common closure with ${\mathcal{M}}_0^{\prime \prime }$. (b) Find a sequence in $\mathcal{B}({\ell }^2)$ that converges in WOT but not in SOT. (Hint: the unilateral shift, as in Example 2.2.) (c) Find a sequence that converges in SOT but not in norm. (Hint: projections onto initial segments.) (d) Verify that on $\mathcal{B}({\mathbb{C}}^n)$ all four topologies (norm, SOT, WOT, σ-weak) coincide.

2. The vN algebra of a self-adjoint operator. For $a=a^{\ast }\in \mathcal{B}(\mathcal{H})$, show: (a) $W^{\ast }(a)=\{a{\}}^{\prime \prime }\cong L^{\infty }(\sigma (a),E_a)$ where $E_a$ is the spectral measure of $a$. (b) $W^{\ast }(a)$ is a maximal abelian vN algebra iff the spectrum of $a$ is multiplicity-free. (c) Find an explicit example of a self-adjoint $a\in M_4(\mathbb{C})$ for which $W^{\ast }(a)$ is not maximal abelian. Compute $W^{\ast }(a)$ and its commutant explicitly. (d) For $a,b\in M_n(\mathbb{C})$ self-adjoint and commuting, compute $\{a,b{\}}^{\prime \prime }$ in terms of the joint spectral measure.

3. Tensor commutant theorem (finite dimensional version). Prove $({\mathcal{A}}_1\otimes {\mathcal{A}}_2{)}^{\prime }={\mathcal{A}}_1^{\prime }\otimes {\mathcal{A}}_2^{\prime }$ for ${\mathcal{A}}_i\subset M_{n_i}(\mathbb{C})$ unital *-subalgebras. (a) First treat the case ${\mathcal{A}}_1={\mathcal{A}}_2=M_2(\mathbb{C})$ explicitly. (b) Use the Wedderburn decomposition for general ${\mathcal{A}}_i$. (c) Connect the result to the structure of bipartite quantum systems: the algebra ${\mathcal{A}}_1\otimes 1$ is the algebra of “Alice’s observables” on ${\mathcal{H}}_1\otimes {\mathcal{H}}_2$.

4. Spectral projections in the algebra. Let $\mathcal{M}$ be a vN algebra and $a=a^{\ast }\in \mathcal{M}$. (a) Show that for any Borel $E\subset \mathbb{R}$, the spectral projection $1_E(a)$ is in $\mathcal{M}$. (b) Show that any bounded measurable function $f(a)$ is in $\mathcal{M}$. (c) Use (b) to show that $\mathcal{M}$ contains the vN algebra of every operator it contains: $W^{\ast }(a)\subset \mathcal{M}$ for every $a\in {\mathcal{M}}_{sa}$.

5. Two projections. Let $p,q\in \mathcal{B}(\mathcal{H})$ be projections. (a) Show $\{p,q{\}}^{\prime \prime }$ is in general non-commutative. (b) For $\mathcal{H}={\mathbb{C}}^2$, find all possible $\{p,q{\}}^{\prime \prime }$ up to unitary equivalence. Separate first the trivial ranks $0$ and $2$. When both projections have rank one, parametrize their relative position by $\mathrm{Tr}(pq)={\cos }^2\theta$. (c) For “generic” $p,q$ in ${\mathbb{C}}^2$ (i.e., $\theta \ne 0,\pi /2$), show $\{p,q{\}}^{\prime \prime }=M_2(\mathbb{C})$.

Starred problems.

6*. Bicommutant in finite dimensions. Give a complete proof of the bicommutant theorem in $M_n(\mathbb{C})$. Use the Wedderburn decomposition (Lemma 4.2) to make the proof entirely elementary. Identify the commutant ${\mathcal{M}}^{\prime }$ explicitly in terms of the Wedderburn data $\{(m_i,k_i)\}$. Verify that ${\mathcal{M}}^{\prime \prime }=\mathcal{M}$ by direct calculation.

7*. Group vN algebras for ${\mathbb{Z}}^n$ and $F_n$. (a) For $G=\mathbb{Z}$, identify $\mathcal{L}(\mathbb{Z})\cong L^{\infty }(\mathbb{T})$ via the Fourier transform. Specifically: show that the unitary $\lambda (1)$ has spectrum $\mathbb{T}$, and identify the spectral measure with the Haar measure on $\mathbb{T}$. (b) For $G={\mathbb{Z}}^n$, identify $\mathcal{L}({\mathbb{Z}}^n)\cong L^{\infty }({\mathbb{T}}^n)$. (c) (Harder.) For $G=F_2$ (the free group on two generators), show that $F_2$ is icc and hence that $\mathcal{L}(F_2)$ is a factor. Week 3 identifies it as type II${}_1$ and explains the classical Murray–von Neumann property-$\Gamma$ argument showing that it is not isomorphic to the hyperfinite II${}_1$ factor.

8*. The σ-weak topology. Show that on $\mathcal{B}(\mathcal{H})$, the σ-weak topology agrees with WOT on norm-bounded sets but is strictly finer in general. For agreement on a ball, approximate an arbitrary trace-class operator in trace norm by finite-rank operators and use $|\mathrm{Tr}((\rho -{\rho }_N)T)|\le \Vert \rho -{\rho }_N{\Vert }_1\Vert T\Vert$. For strictness, show that WOT-continuous linear functionals are finite sums of vector functionals, hence correspond to finite-rank trace-class operators, whereas $T\mapsto \mathrm{Tr}(\rho T)$ is σ-weakly continuous for every trace-class $\rho$; choose $\rho$ of infinite rank. Explain why a WOT-convergent sequence cannot serve as the counterexample: uniform boundedness makes such a sequence norm-bounded.

9** (Optional, hard). Read Sakai’s theorem (Theorem 6.2) and reproduce its proof in the case of $\mathcal{B}(\mathcal{H})$: show that $\mathcal{B}(\mathcal{H})$ is the dual Banach space of $\mathcal{T}(\mathcal{H})$ (trace-class operators with the trace-norm). Identify the σ-weak topology as the weak-* topology induced by this duality.

Self-study answer checkpoints

Use these to test the core problems after completing them. Starred problems require the cited structural results or reading and are intentionally not reduced to answer keys here.

1. Operator topologies. For a unital $\ast$-algebra, both WOT and SOT closures equal the bicommutant. The unilateral shifts satisfy $S^n\to 0$ weakly but not strongly. The initial-segment projections $P_n$ satisfy $P_n\to 1$ strongly while $\Vert 1-P_n\Vert =1$. On $M_n(\mathbb{C})$ all four topologies coincide.

2. One self-adjoint generator. The correct model is bounded Borel functional calculus modulo $E_a$-null functions:

   $$W^{\ast }(a)=\{f(a):f\text{bounded Borel}\}\cong L^{\infty }(\sigma (a),E_a).$$

   It is maximal abelian exactly at spectral multiplicity one. For example, $a=\mathrm{diag}(0,0,1,1)$ has $W^{\ast }(a)=\{\alpha I_2\oplus \beta I_2\}$ and commutant $M_2\oplus M_2$, so it is not maximal abelian. For commuting matrices, the joint eigenspaces—and only their degeneracies—determine $\{a,b{\}}^{\prime \prime }$.

3. Tensor commutants. Wedderburn decomposition reduces the calculation blockwise and yields

   $$({\mathcal{A}}_1\otimes {\mathcal{A}}_2{)}^{\prime }={\mathcal{A}}_1^{\prime }\otimes {\mathcal{A}}_2^{\prime }.$$

   In the full $M_2\otimes M_2=M_4$ case the commutant is $\mathbb{C}1$. For Alice’s algebra ${\mathcal{A}}_1\otimes 1$, the commuting algebra is ${\mathcal{A}}_1^{\prime }\otimes M_{n_2}$.

4. Spectral projections. Approximate $1_E$ pointwise by a uniformly bounded net of Borel/simple or continuous functions and use the spectral theorem: $1_E(a)\in \mathcal{M}$. Bounded Borel functional calculus then gives $f(a)\in \mathcal{M}$, and therefore $W^{\ast }(a)\subset \mathcal{M}$.

5. Two projections in ${\mathbb{C}}^2$. A generic pair of distinct, nonorthogonal rank-one projections does not commute and generates $M_2$. Up to unitary equivalence, the alternatives are: only scalars if both projections are trivial; a diagonal copy $\mathbb{C}\oplus \mathbb{C}$ if one nontrivial projection suffices or two rank-one projections commute; and $M_2$ when $0<\mathrm{Tr}(pq)<1$.

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Notes prepared for the algebraic-QFT course, Semester I Block A. Last revised 2026-09-29.