Measurable fields of von Neumann algebras have measurable commutants and centers

Discard a Borel null set where the defining generation identity fails. The dimension strata $X_p=\{x:\dim H_x=p\}$, $p\in \{1,2,\dots \}\cup \{\mathrm{\infty }\}$, are Borel by [F3], their union with the zero stratum is $X$, and on the zero stratum $H_x=\{0\}$, so $M_x=M_x^{\mathrm{\prime }}=Z(M_x)=\{0\}$ and all claims are trivial; on a fixed stratum $X_p$ we may therefore use the unitaries $U_x:H_x\to K_p$ onto the fixed separable model of [F3] and the transported algebras $M_x^t:=U_x{M}_x{U}_x^{\ast }$.

Let $B$ be the WOT unit ball of $\mathcal{B}(K_p)$ with the metric $d(T,S):={\sum }_{a,b}{2}^{-(a+b)}\min (1,\mathrm{\mid }⟨(T-S)f_a,f_b⟩\mathrm{\mid })$ attached to a fixed orthonormal basis $(f_a)$ of $K_p$; the basis-coordinate topology agrees with WOT by [F6]. To prove compactness, form the compact product of closed unit discs indexed by $(a,b)$. The displayed weighted coefficient metric induces its product topology: finitely many coordinates control each finite head, and the summable weights uniformly control the tail. In it impose $\mathrm{\mid }{\sum }_{a,b}{c}_{ab}{z}_a\overline{w_b}\mathrm{\mid }\le \mathrm{\parallel }z\mathrm{\parallel }\mathrm{\parallel }w\mathrm{\parallel }$ for all finite rational-complex coordinate vectors $z,w$. These are closed conditions, so their solution set is compact. Scalar continuity extends the inequalities to all finite complex coordinate vectors; the resulting bounded sesquilinear form extends by density to $K_p$. Riesz representation, with the first-variable-linear convention, represents it uniquely as $⟨Tz,w⟩$ for a contraction $T$. Conversely every contraction satisfies the conditions, so this closed coordinate set is exactly $B$. Thus $B$ with the displayed metric is compact. Enumerate only the finite matrices with rational-complex entries and operator norm at most $1$, extended by zero on the remaining coordinates. This is a countable subset of $B$ and is WOT-dense: finite-coordinate compressions $P_{n}T{P}_n$ of a contraction $T$ converge strongly to $T$; for any positive rational $\epsilon <1$, the finite matrix $(1-\epsilon )P_{n}T{P}_n$ has norm at most $1-\epsilon$ and can be approximated in finite-dimensional operator norm by rational-complex matrices within any tolerance less than $\epsilon$, all still of norm at most $1$. Taking the compression size to infinity and the shrinkage and tolerances to zero proves the asserted WOT density.

The direct integral $\mathcal{M}$ is a unital $\ast$-subalgebra of $\mathcal{B}(\mathcal{H})$ containing the diagonal algebra $\mathcal{D}$: sums, products and adjoints of induced operators are induced by the pointwise sums, products and adjoints of essentially bounded weakly measurable fields by [F2], the identity is induced by the constant field $I_{H_x}$, and $\mathcal{D}\subseteq \mathcal{M}$ by [F1].

Explicitly adjoin adjoints to the defining sequence: set $R_{2k-1}(x)=T_x^{(k)}$ and $R_{2k}(x)=(T_x^{(k)}{)}^{\ast }$, so $M_x=W^{\ast }(R_j(x):j\ge 1)$ and the family is adjoint-closed. The transported generators $S_j(x):=U_x{R}_j(x)U_x^{\ast }$ are weakly measurable by [F2,F3]. Their norms are Borel, since they are the suprema of their norms on a fixed countable dense subset of the unit sphere in the constant-space model; the latter norms are Borel limits of finite coefficient square sums. Put $R_j^b(x):=R_j(x)\mathrm{/}(1+\mathrm{\parallel }{R}_j(x)\mathrm{\parallel })$ on the original fibres and $S_j^b(x):=U_x{R}_j^b(x)U_x^{\ast }=S_j(x)\mathrm{/}(1+\mathrm{\parallel }{S}_j(x)\mathrm{\parallel })$ on $K_p$. The original fields are weakly measurable and bounded by $1$ on the countable union of strata, with value $0$ on the zero stratum. Each normalized family generates its corresponding fibre algebra, since normalization multiplies each generator by a nonzero scalar. The normalized family remains adjoint-closed because an operator and its adjoint have the same norm. Therefore commuting with every $S_j^b(x)$ is equivalent to commuting with $M_x^t$: it gives commutation with the generated unital $\ast$-algebra and then with its WOT closure, since multiplication by a fixed bounded operator is WOT-continuous. The assignment $x\mapsto (S_1^b(x),S_2^b(x),\dots )$ is Borel into the product WOT balls by its measurable coordinates.

We claim $(\mathcal{M}{)}^{\mathrm{\prime }}\subseteq {\int }_X^{\oplus }{M}_x^{\mathrm{\prime }}d\mu (x)$. Let $T\in (\mathcal{M}{)}^{\mathrm{\prime }}$; since $\mathcal{D}\subseteq \mathcal{M}$, the operator $T$ commutes with $\mathcal{D}$ and hence is decomposable, $T={\int }_X^{\oplus }{T}_{x}d\mu (x)$ for a weakly measurable essentially bounded field $(T_x)$, by [F2]. For every $k$ the operator ${\int }_X^{\oplus }{R}_k^b(x)d\mu (x)$ belongs to $\mathcal{M}$, so $T$ commutes with it; by [F2] the field $x\mapsto [T_x,R_k^b(x)]$ induces the zero operator, and a decomposable operator vanishes exactly when its field vanishes almost everywhere, as its coefficient integrals against a countable fundamental family of sections all vanish. Hence, on one conull set depending on $k$, the fibre $T_x$ commutes with $R_k^b(x)$; intersecting the countably many conull sets gives one conull set on which $T_x$ commutes with every member of the adjoint-closed normalized generating family of step 2.1, so $T_x\in M_x^{\mathrm{\prime }}$ almost everywhere, and $T\in {\int }_X^{\oplus }{M}_x^{\mathrm{\prime }}d\mu (x)$. The reverse inclusion is pointwise commutation.

Define $g(x,T):={\sum }_{k,a,b}{w}_{k,a,b}\min (1,\mathrm{\mid }⟨(T{S}_k^b(x)-S_k^b(x)T)f_a,f_b⟩\mathrm{\mid })$ with positive summable weights $w_{k,a,b}$. For fixed $T$ the map $x\mapsto g(x,T)$ is measurable, a countable sum of measurable functions by step 1.2; for fixed $x$ the map $T\mapsto g(x,T)$ is continuous, being the uniform limit of the weighted partial sums of continuous functions; and $g(x,T)=0$ exactly when $T$ commutes with every $S_k^b(x)$, that is, exactly when $T\in (M_x^t{)}^{\mathrm{\prime }}$ and $\mathrm{\parallel }T\mathrm{\parallel }\le 1$, by step 2.1. The zero sets $C_x=(M_x^t{)}_1^{\mathrm{\prime }}$ are nonempty, since the identity belongs to them, and compact.

Apply [F4] on the standard Borel sigma-finite space $X_p$ to the function $g$, and reindex its selectors by $B_j=s_{j-1}$ for $j\ge 1$. This yields measurable maps $B_j:X_p\to B$ with $B_j(x)\in (M_x^t{)}_1^{\mathrm{\prime }}$ for all $j\ge 1$ such that $(B_j(x){)}_{j\ge 1}$ is WOT-dense in $(M_x^t{)}_1^{\mathrm{\prime }}$ for every $x$. Each $B_j$ is a weakly measurable operator field, since its Borel matrix coefficients in the basis $(f_a)$ are obtained by composing the coefficient functionals with the measurable map $B_j$; hence $(M_x^t{)}^{\mathrm{\prime }}=W^{\ast }(B_j(x):j\ge 1)$ is a measurable field of von Neumann algebras in the sense of [F1].

Repeating steps 3.2–4.1 for the simultaneous commutator equations of the fields $S_k^b$ and $B_j$ yields measurable dense sections of the unit ball of the centre $Z(M_x^t)=(M_x^t)\cap (M_x^t{)}^{\mathrm{\prime }}$; repeating them for the commutator equations of the fields $B_j$ alone yields measurable dense sections of the unit ball of $M_x^t$, because the commutant of the WOT-closed unital algebra generated by the $B_j$ is exactly $\{T:T{B}_j=B_{j}T\text{ for all }j\}$; and by [F5] the countably many selections so obtained, together with the Gram-Schmidt sections of [F3], can be combined on one common conull Borel subset of $X_p$.

Transporting back by the unitaries $U_x$, and taking the union over the countably many dimension strata inside one common conull set, we obtain the promised WOT-dense countable measurable sections of the unit balls of $M_x$, $M_x^{\mathrm{\prime }}$ and $Z(M_x)$ on a common conull Borel stratum, and the fields $x\mapsto M_x^{\mathrm{\prime }}$, $x\mapsto Z(M_x)$ satisfy the measurability condition of [F1] through those sections.

Therefore $(\mathcal{M}{)}^{\mathrm{\prime }}={\int }_X^{\oplus }{M}_x^{\mathrm{\prime }}d\mu (x)$, and, since $x\mapsto M_x^{\mathrm{\prime }}$ is again a measurable field of von Neumann algebras by step 6.1, the same identity applies to it: $(\mathcal{M}{)}^{\mathrm{\prime }\mathrm{\prime }}={\int }_X^{\oplus }(M_x^{\mathrm{\prime }}{)}^{\mathrm{\prime }}d\mu (x)={\int }_X^{\oplus }{M}_{x}d\mu (x)=\mathcal{M}$, where the middle equality is the fibre double commutant theorem of [F6] applied to each $M_x$. Hence $\mathcal{M}$ is a WOT-closed unital $\ast$-subalgebra of $\mathcal{B}(\mathcal{H})$, that is, a concrete von Neumann algebra, with commutant ${\mathcal{M}}^{\mathrm{\prime }}={\int }_X^{\oplus }{M}_x^{\mathrm{\prime }}d\mu (x)$.

For the centre: by [F2] the intersection $\mathcal{M}\cap {\mathcal{M}}^{\mathrm{\prime }}$ consists exactly of those operators whose fibres lie in $M_x\cap M_x^{\mathrm{\prime }}$ almost everywhere, because an operator in the intersection has two decomposable representatives with fibres in $M_x$ and in $M_x^{\mathrm{\prime }}$ respectively, and decomposable representatives are unique almost everywhere. Hence $Z(\mathcal{M})={\int }_X^{\oplus }Z(M_x)d\mu (x)$.

Finally let $(N_x)$ be a measurable field of unital von Neumann algebras with ${\int }_X^{\oplus }{N}_{x}d\mu (x)={\int }_X^{\oplus }{M}_{x}d\mu (x)$. For each $k$, the operator ${\int }_X^{\oplus }{R}_k^b(x)d\mu (x)$ belongs to ${\int }_X^{\oplus }{N}_{x}d\mu (x)$, so by the almost-everywhere uniqueness of decomposable representatives its field agrees almost everywhere with an $N_x$-valued essentially bounded weakly measurable field; thus $R_k^b(x)\in N_x$ for almost every $x$. Intersecting the countably many conull sets and taking weak-operator closures of the generated algebras gives $M_x\subseteq N_x$ almost everywhere; the symmetric argument gives $N_x\subseteq M_x$ almost everywhere, so the two fields coincide almost everywhere.