# The central decomposition and the types of the fibres

*Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).*

Let \(Q\) be a von Neumann algebra on a separable Hilbert space. Its centre is an abelian von Neumann algebra. This lesson shows that the centre can be written as the diagonal algebra of a direct integral over a standard Borel space (Theorem 2.1), so that \(Q\) becomes a direct integral \(\int^\oplus M(t)\,d\mu(t)\) of factors (Theorem 4.1). This is the *central decomposition*. It is unique up to a Borel isomorphism of the bases, defined outside null sets, and a measurable field of unitaries (Theorem 4.4). Many properties of \(Q\) can then be read off from the fibres \(M(t)\). Countable generating sets, cyclic and separating sets of vectors, and tracial vector functionals pass to almost every fibre (Theorems 4.1 and 4.2). The type passes in both directions: \(Q\) is finite, properly infinite, semifinite, of type I, II, II\(_1\), II\(_\infty\) or III exactly when almost every fibre is (Theorem 5.3).

The tools are a countable algebra of operators whose fibres are dense in almost every fibre algebra (Lemma 3.1), sections whose values are total in almost every fibre (Lemma 3.2), and the fibres of a sum of vector functionals (Lemmas 3.3 and 3.4). Most statements about types are proved with traces and the type decomposition of \(Q\). Type III needs one more ingredient: a measurable choice of finite projections and of traces on them in the fibres (Lemma 5.1).

The decomposition goes back to von Neumann's reduction theory. Statements of the central decomposition and of its compatibility with types are in [Blackadar, III.1.6.3–III.1.6.4 and III.5.1.15]. Within this programme, the existence and uniqueness of the decomposition are also proved, by a different route (measurable selectors for countably many operator equations), in the lesson Central decomposition from countable operator equations of the course on modular theory, by GPT-6.1 Sol (OpenAI). The type III argument below follows an idea of Takesaki, a measurable selection of finite projections with faithful traces, and writes out the Borel conditions explicitly.

We assume the lessons Direct integrals of von Neumann algebras, Unbounded decomposable operators and measurable spectral decomposition (only its Theorem 6.1), Projections and types of von Neumann algebras, Traces on von Neumann algebras and Polish spaces and standard Borel spaces, and the lessons they rest on. Every fact we take from them is stated in Section 1.

## 1. Conventions and results used

Hilbert spaces are complex, and inner products are linear in the first variable. For a Hilbert space \(K\), \(B(K)\) is the algebra of bounded operators, \(S'\) is the commutant of a set \(S\subseteq B(K)\), a *von Neumann algebra* is a \(*\)-subalgebra \(M\) with \(M''=M\), and the von Neumann algebra *generated* by a set \(S\) is \((S\cup S^*)''\). The centre of \(M\) is \(M\cap M'\), and \(M\) is a *factor* if its centre is \(\mathbb C1\). On the zero space, \(B(0)=\{0\}\) is a factor.

*Projections and types.* For projections \(e,f\) of a von Neumann algebra \(M\), \(e\sim f\) means \(e=v^*v\) and \(f=vv^*\) for some \(v\in M\), and \(e\precsim f\) means \(e\sim f_1\le f\) for some projection \(f_1\in M\). The central support \(c(e)\) is the least central projection majorizing \(e\). A projection \(e\) is *finite* if \(e\sim f\le e\) implies \(f=e\); *properly infinite* if every nonzero central cut \(ze\) is infinite; *purely infinite* if \(0\) is its only finite subprojection; *abelian* if \(eMe\) is commutative. The algebra \(M\) has one of these properties if the projection \(1\) has it. The types I, II, II\(_1\), II\(_\infty\) and III and the word *semifinite* are as in Projections and types of von Neumann algebras, Definition 7.1: type I means that below each nonzero central projection there is a nonzero abelian projection; type II, that \(0\) is the only abelian projection while below each nonzero central projection there is a nonzero finite projection; type III means purely infinite; type II\(_1\) means type II and finite; type II\(_\infty\) means type II with \(0\) as the only finite central projection; semifinite means that there is no nonzero central summand of type III. By these definitions the zero algebra has every one of these properties. A nonzero factor is of exactly one of the types I, II\(_1\), II\(_\infty\), III; it is either finite or properly infinite, since its only nonzero central projection is \(1\); and it is semifinite exactly when it is not of type III.

*Vector functionals.* For a sequence \(\zeta=(\zeta_n)\) in \(K\) with \(\sum_n\|\zeta_n\|^2<\infty\), put \(\omega_\zeta(x)=\sum_n\langle x\zeta_n,\zeta_n\rangle\) for \(x\in B(K)\). It is a positive linear functional, and it is \(\sigma\)-weakly continuous by the definition of the \(\sigma\)-weak topology (The double commutant theorem, Definition 1.1). Let \(M\) be a von Neumann algebra on \(K\) and \(p\in M\) a projection. We say that \(\omega_\zeta\) is *tracial on \(pMp\)* if \(\omega_\zeta(pxpyp)=\omega_\zeta(pypxp)\) for all \(x,y\in M\), and *faithful on \(pMp\)* if \(x\in M\) and \(\omega_\zeta(px^*xp)=0\) imply \(xp=0\).

*Direct integrals.* \((\Gamma,\Sigma,\mu)\) is a \(\sigma\)-finite measure space and \((H(\gamma))\) a measurable field of Hilbert spaces over it, with direct integral \(\mathcal H=\int^\oplus_\Gamma H(\gamma)\,d\mu(\gamma)\). A *null set* is a set \(N\in\Sigma\) with \(\mu(N)=0\), and *almost every* \(\gamma\) means every \(\gamma\) outside a null set. We use one letter for a vector of \(\mathcal H\) and for a chosen square-integrable measurable section that represents it. For a bounded measurable \(f\), \(m_f\) is multiplication by \(f\). For an essentially bounded measurable operator field \(x\), \(\int^\oplus x\) is the decomposable operator \((\int^\oplus x\,\xi)(\gamma)=x(\gamma)\xi(\gamma)\). The diagonal algebra is \(\mathcal A=\{m_f\}\), and the decomposable algebra \(\mathcal D\) is the set of all \(\int^\oplus x\). We write \(n(\gamma)=\dim H(\gamma)\) and \(\Gamma_d=\{n=d\}\) for \(d\in\{0,1,2,\ldots,\infty\}\), and \(\ell^2_d\) is \(\mathbb C^d\), or \(\ell^2(\mathbb N)\) if \(d=\infty\), with standard basis \((\varepsilon_i)\).

**Fact 1.1** (measurable fields and decomposable operators). From Measurable fields of Hilbert spaces and their direct integrals and Decomposable operators and the diagonal algebra:

- (a) inner products and norms of measurable sections are measurable functions; \(f\xi\) is measurable for measurable \(f\) and measurable \(\xi\); a pointwise weak limit of measurable sections is measurable; measurable sections can be glued along a countable measurable partition (Lemma 2.2);
- (b) for a sequence \((\eta_j)\) of measurable sections, the orthogonal projections \(P(\gamma)\) onto the closed linear span of \(\{\eta_j(\gamma)\}\) form a measurable operator field (Proposition 4.1(1));
- (c) the function \(n\) is measurable, and there are measurable sections \(e_k\), \(k\ge1\), with \(e_k(\gamma)\ne0\) exactly when \(k\le n(\gamma)\), whose nonzero values form an orthonormal basis of \(H(\gamma)\); for \(\gamma\in\Gamma_d\), \(U(\gamma)v=\sum_{k\le d}\langle v,e_k(\gamma)\rangle\varepsilon_k\) is a unitary of \(H(\gamma)\) onto \(\ell^2_d\) (Theorems 3.1 and 5.1); on the constant field \(\ell^2_d\) over \(\Gamma_d\), the measurable sections are the maps with measurable coordinates (Example 11.2), and a family of bounded operators is a measurable operator field exactly when its matrix entries \(\gamma\mapsto\langle y(\gamma)\varepsilon_i,\varepsilon_j\rangle\) are measurable (Proposition 8.1);
- (d) a family of bounded operators between two measurable fields is a *measurable operator field* if it maps measurable sections to measurable sections, and it suffices to check this on one fundamental sequence; sums, composites and adjoints of measurable operator fields, and their products with measurable functions, are measurable, and so is the norm function \(\gamma\mapsto\|x(\gamma)\|\) (Definition 6.1 and Theorem 6.2); in particular \(U\) and \(U^*\) are measurable operator fields between the field \((H(\gamma))_{\gamma\in\Gamma_d}\) and the constant field \(\ell^2_d\), because \(U\) maps measurable sections to sections with measurable coordinates and \(U^*\varepsilon_k=e_k\) for \(k\le d\);
- (e) for an essentially bounded measurable operator field \(x\), \(\int^\oplus x\) is bounded, with norm \(\operatorname*{ess\,sup}_\gamma\|x(\gamma)\|\); the map \(x\mapsto\int^\oplus x\) is linear and multiplicative, and \((\int^\oplus x)^*=\int^\oplus x^*\); two fields define the same operator exactly when they agree almost everywhere; every \(\int^\oplus x\) commutes with every \(m_f\) (Theorem 10.1); if \(\Sigma\) is the Borel \(\sigma\)-algebra of a standard Borel space, \(\mathcal H\) is separable (Theorem 9.1);
- (f) \(\mathcal A\) and \(\mathcal D\) are von Neumann algebras, \(\mathcal A'=\mathcal D\) and \(\mathcal D'=\mathcal A\); every \(T\in\mathcal A'\) is \(\int^\oplus x\) for a measurable operator field \(x\) with \(\|x(\gamma)\|\le\|T\|\) for every \(\gamma\); and \(m_f=0\) exactly when \(f=0\) almost everywhere on \(\{n\ge1\}\) (Theorems 5.1 and 7.1, Proposition 2.2);
- (g) let \((\xi_j)\) be sections of a family of Hilbert spaces \((H(\gamma))\) over \((\Gamma,\Sigma)\) whose Gram functions \(\gamma\mapsto\langle\xi_j(\gamma),\xi_i(\gamma)\rangle\) are measurable and whose values are total in every \(H(\gamma)\); then the sections \(\eta\) for which every \(\gamma\mapsto\langle\eta(\gamma),\xi_j(\gamma)\rangle\) is measurable form a measurable field, the only one containing every \(\xi_j\); and in any measurable field, a section is measurable as soon as its inner products with the members of one fundamental sequence are measurable (Theorem 3.1(3), (4));
- (h) for two measurable fields over the same base, a bounded operator \(T\) between their direct integrals with \(Tm_f=m_fT\) for every \(f\) is \(\int^\oplus x\) for a measurable field \(x\) of bounded operators with \(\|x(\gamma)\|\le\|T\|\) for every \(\gamma\) (Proposition 6.1); if \(T\) is unitary, \(x(\gamma)\) is unitary for almost every \(\gamma\) (Exercise 10.3 and its solution);
- (i) *Radon–Nikodym.* If \(\nu\) and \(\lambda\) are \(\sigma\)-finite measures on \(\Sigma\) with the same null sets, then \(\nu(B)=\int_B\rho\,d\lambda\) for a measurable \(\rho:\Gamma\to(0,\infty)\). For finite measures this is in Decomposable operators and the diagonal algebra, Results used from other lessons; in general, split \(\Gamma\) into countably many measurable sets of finite \(\nu\)- and \(\lambda\)-measure, apply the finite case on each, and replace \(\rho\) by \(1\) on the set \(\{\rho=0\}\), which is \(\nu\)-null and hence \(\lambda\)-null.

**Fact 1.2** (direct integrals of von Neumann algebras). From Direct integrals of von Neumann algebras. A *field of von Neumann algebras* is a family \(M=(M(\gamma))\) with \(M(\gamma)\) a von Neumann algebra on \(H(\gamma)\). It is *measurable* if there are measurable operator fields \(x_j\), \(j\ge1\), such that \(M(\gamma)\) is generated by \(\{x_j(\gamma)\}\) for almost every \(\gamma\). Fields that agree almost everywhere are *equivalent*, and a field equivalent to a measurable field is measurable (Definition 1.1 and Remark 1.4(2)). We write \(M'=(M(\gamma)')\) and \(\int^\oplus M=\{\int^\oplus x: x\text{ essentially bounded and measurable},\ x(\gamma)\in M(\gamma)\text{ for almost every }\gamma\}\).

- (a) If \(M\) is measurable, so is \(M'\). For a measurable \(M\) there are measurable operator fields \(y_k\), \(k\ge1\), with \(\|y_k(\gamma)\|\le1\) for every \(\gamma\), such that for almost every \(\gamma\) the \(y_k(\gamma)\) lie in \(M(\gamma)\) and are \(\sigma\)-weakly dense in its unit ball (Proposition 1.3(2), (3)).
- (b) Every \(T\in\int^\oplus M\) equals \(\int^\oplus x\) for a measurable operator field \(x\) with \(x(\gamma)\in M(\gamma)\) and \(\|x(\gamma)\|\le\|T\|\) for every \(\gamma\); every essentially bounded field representing \(T\) has its values in \(M(\gamma)\) almost everywhere; \(\int^\oplus M\) is a \(*\)-algebra containing \(\mathcal A\); and \(M_1(\gamma)\subseteq M_2(\gamma)\) almost everywhere implies \(\int^\oplus M_1\subseteq\int^\oplus M_2\) (Proposition 2.2(1)–(3)).
- (c) For measurable \(M\): \((\int^\oplus M)'=\int^\oplus M'\), and \(\int^\oplus M\) is a von Neumann algebra whose centre contains \(\mathcal A\) (Theorem 3.2).
- (d) For measurable \(M\) and any field \(L\), \(\int^\oplus M\subseteq\int^\oplus L\) exactly when \(M(\gamma)\subseteq L(\gamma)\) for almost every \(\gamma\); two measurable fields with the same direct integral are equivalent (Theorem 4.1).
- (e) For measurable \(M\), the centre of \(\int^\oplus M\) equals \(\mathcal A\) exactly when \(M(\gamma)\) is a factor for almost every \(\gamma\) (Theorem 4.3).
- (f) Let \(S\subseteq\mathcal D\) be countable, choose essentially bounded measurable fields \(\beta_s\) with \(s=\int^\oplus\beta_s\), and let \(M_S(\gamma)\) be the von Neumann algebra generated by \(\{\beta_s(\gamma):s\in S\}\). Then \(M_S\) is measurable, and the von Neumann algebra generated by \(\mathcal A\cup S\) is \(\int^\oplus M_S\) (Lemma 5.1).
- (g) A von Neumann algebra \(\mathcal B\) on \(\mathcal H\) equals \(\int^\oplus M\) for a measurable field \(M\) exactly when \(\mathcal A\subseteq\mathcal B\subseteq\mathcal D\) (Theorem 5.3).
- (h) A von Neumann algebra on a separable Hilbert space is generated by a countable set (Remark 1.4(1)). For a separable \(K\), the closed unit ball of \(B(K)\) with the \(\sigma\)-weak topology is a compact metrizable space (The Effros Borel structure, Proposition 3.2(7) and Lemma 8.1(1), with \(E=B(K)_*\)).
- (i) *Unitary transfer.* If \(v\) is a measurable field of unitaries \(v(\gamma):H(\gamma)\to K(\gamma)\) between two measurable fields over the same base and \(V=\int^\oplus v\), then \(V(\int^\oplus M)V^*=\int^\oplus vMv^*\), where \(vMv^*=(v(\gamma)M(\gamma)v(\gamma)^*)\), and \(vMv^*\) is measurable if \(M\) is (Proposition 2.3).

**Fact 1.3** (diagonalizing a calculus). From Unbounded decomposable operators and measurable spectral decomposition, Theorem 6.1. Let \((Z,\mathcal Z,\lambda)\) be a \(\sigma\)-finite measure space, \(B_b(Z)\) the algebra of bounded measurable functions, \(K\) a Hilbert space and \(\Pi:B_b(Z)\to B(K)\) a map such that (i) \(\Pi\) is linear and multiplicative, \(\Pi(\bar F)=\Pi(F)^*\) and \(\Pi(1)=1\); (ii) \(F_n\to F\) pointwise with \(\sup_n\sup_Z|F_n|<\infty\) implies \(\Pi(F_n)\to\Pi(F)\) strongly; (iii) \(\Pi(F)=0\) when \(F=0\) \(\lambda\)-almost everywhere; (iv) for some sequence \((\eta_k)\) in \(K\), the vectors \(\Pi(F)\eta_k\) span a dense subspace. Then there are sets \(Z_k\in\mathcal Z\) and a unitary \(W\) from \(K\) onto the direct integral over \((Z,\mathcal Z,\lambda)\) of the measurable field \(H(z)=\overline{\operatorname{span}}\{\varepsilon_k:z\in Z_k\}\subseteq\ell^2\), whose measurable sections are the sections with measurable coordinates, such that \(W\Pi(F)W^*=m_F\) for every \(F\in B_b(Z)\).

**Fact 1.4** (one generator and its Borel calculus).

- (a) An abelian von Neumann algebra on a separable Hilbert space is generated by one self-adjoint element (Abelian operator algebras, Theorem 8.2).
- (b) Let \(a\in B(K)\) be self-adjoint with spectrum \(S\). For every bounded Borel function \(F\) on \(\mathbb R\) there is an operator \(F(a)\in B(K)\) such that \(F\mapsto F(a)\) is linear and multiplicative, \(\bar F(a)=F(a)^*\) and \(1(a)=1\); for continuous \(F\) it is the continuous functional calculus of \(F|_S\); \(\langle F(a)\xi,\xi\rangle=\int F\,d\mu_\xi\) and \(\|F(a)\xi\|^2=\int|F|^2\,d\mu_\xi\) for a finite positive Borel measure \(\mu_\xi\) on \(\mathbb R\) carried by \(S\), with \(\mu_\xi(\mathbb R)=\|\xi\|^2\); if \(F_n\to F\) pointwise and boundedly, then \(F_n(a)\to F(a)\) strongly; and \(F(a)\) commutes with every operator that commutes with \(a\) (The spectral theorem for bounded self-adjoint operators, Theorem 3.1, where \(\mu_\xi\) is a measure on \(S\), extended here by \(0\)).

**Fact 1.5** (topologies, density and normal functionals). From The double commutant theorem:

- (a) for fixed \(a,b\in B(K)\), the map \(x\mapsto axb\) is continuous for the strong and for the \(\sigma\)-weak topology (Lemma 1.2(c));
- (b) a \(*\)-subalgebra of \(B(K)\) that contains \(1\) is dense in its bicommutant for the strong and for the \(\sigma\)-weak topology (Theorem 4.4(2), (4));
- (c) the projection onto a closed subspace that is invariant under a self-adjoint set \(S\) lies in \(S'\) (Proposition 2.1(5)); a set of vectors is separating for a von Neumann algebra \(M\) exactly when it is cyclic for \(M'\) (Proposition 9.2(3));
- (d) for a projection \(e\) of a von Neumann algebra \(M\) on \(K\), the restrictions to \(eK\) of the operators \(exe\), \(x\in M\), form a von Neumann algebra \(M_e\) on \(eK\), whose commutant consists of the restrictions to \(eK\) of the elements of \(M'\), and whose centre consists of the restrictions of the \(ce\), \(c\) central in \(M\) (Theorem 5.8);
- (e) every \(\sigma\)-weakly continuous positive linear functional \(\varphi\) on a von Neumann algebra \(M\subseteq B(K)\) is \(\omega_\zeta|_M\) for a sequence \(\zeta\) in \(K\) with \(\sum_n\|\zeta_n\|^2<\infty\) (Theorem 10.1).

**Fact 1.6** (projections and types). From Projections and types of von Neumann algebras. Let \(M\) be a von Neumann algebra.

- (a) A subprojection of a finite projection is finite, and a projection equivalent to an abelian projection is abelian. Abelian projections are finite. A projection \(e\) is finite, or abelian, exactly when the algebra \(eMe\) is finite, or commutative. If \(e\) is abelian, then \(eMe=Ze\), where \(Z\) is the centre; so in a factor a nonzero abelian projection \(e\) has \(eMe=\mathbb Ce\), and its only subprojections are \(0\) and \(e\) (Lemma 6.2).
- (b) *Comparison.* If \(M\) is a factor, then for projections \(e,f\) exactly one of \(e\prec f\), \(e\sim f\), \(f\prec e\) holds, where \(e\prec f\) means \(e\precsim f\) and \(e\not\sim f\) (Theorem 5.5).
- (c) *Type decomposition.* There are unique mutually orthogonal central projections \(z_{\rm I},z_{{\rm II}_1},z_{{\rm II}_\infty},z_{\rm III}\) with sum \(1\) such that \(Mz_T\) is of type \(T\) for each \(T\). Moreover \(1=e_{\rm f}+e_{\rm pi}\) for centrally orthogonal projections \(e_{\rm f}\) finite and \(e_{\rm pi}\) properly infinite (Theorem 7.2). For a central projection \(z\), the projections of the central summand \(Mz\) on \(zK\) are the projections of \(M\) below \(z\), its central projections are the central projections of \(M\) below \(z\), and equivalence among them is the same as in \(M\); each type passes to central summands (the remark after Definition 7.1 there).
- (d) *Good projections.* In a type I algebra some abelian projection has central support \(1\); in a semifinite algebra some finite projection has central support \(1\) (Lemma 7.4).
- (e) *Halving.* A properly infinite algebra contains a projection \(e\) with \(e\sim1-e\sim1\) (Proposition 13.4).

**Fact 1.7** (traces). From Traces on von Neumann algebras:

- (a) a finite von Neumann algebra whose centre is \(\sigma\)-finite has a faithful normal tracial state \(\tau\): a positive linear functional with \(\tau(1)=1\), \(\tau(xy)=\tau(yx)\) for all \(x,y\), \(\tau(x^*x)>0\) for \(x\ne0\), which is \(\sigma\)-weakly continuous (Lemma 5.11 and Corollary 5.12(1), built on Theorem 5.2, with the extension of finite traces in Proposition 2.2(6) and the identification of normal positive functionals with \(\sigma\)-weakly continuous ones in the conventions there);
- (b) *division of the unit*: if \(M\) is of type II\(_1\) and \(m\ge1\), then \(1=e_1+\dots+e_m\) with mutually orthogonal, mutually equivalent projections \(e_j\) (Corollary 5.18).

A von Neumann algebra on a separable Hilbert space is \(\sigma\)-finite: unit vectors in the ranges of mutually orthogonal nonzero projections are orthonormal, so there are countably many such projections. In particular, every von Neumann algebra on a separable space has a \(\sigma\)-finite centre.

**Fact 1.8** (Souslin sets and measurable sections). From Polish spaces and standard Borel spaces. A set \(E\) is *\(\mu\)-measurable* if \(B_1\subseteq E\subseteq B_2\) for Borel sets with \(\mu(B_2\setminus B_1)=0\) (Section 6).

- (a) A Borel subset of a Polish space, with the relative topology, is a Souslin space (Corollary 2.6). Countable products of Polish spaces are Polish, and the Borel sets of a product of countably many second countable spaces are generated by the coordinate maps ((F1) and (F2) there).
- (b) If \(f\) is an injective Borel map from a standard Borel space into a separable metrizable space, its image is Borel and \(f\) is a Borel isomorphism onto it (Theorem 4.3(5)).
- (c) *Measurable sections.* Let \(X\) be a Souslin space, \(Y\) a separable metrizable space and \(f:X\to Y\) a Borel map. Then \(f(X)\) is a Souslin set, and there is \(\varphi:f(X)\to X\) with \(f(\varphi(y))=y\) such that \(\varphi^{-1}(B)\) is \(\mu\)-measurable for every Borel \(B\subseteq X\) and every \(\sigma\)-finite measure \(\mu\) on \(Y\); and Souslin sets are \(\mu\)-measurable (Theorems 6.2 and 7.7).

## 2. Abelian von Neumann algebras on separable spaces

**Theorem 2.1.** Let \(\mathcal Z\) be an abelian von Neumann algebra on a separable Hilbert space \(K\ne0\). There are a Borel probability measure \(\mu\) on \(\mathbb R\), a measurable field of Hilbert spaces \((H(t))_{t\in\mathbb R}\) over \(\mathbb R\) with its Borel sets and \(\mu\), and a unitary \(W\) from \(K\) onto \(\mathcal H=\int^\oplus_{\mathbb R}H(t)\,d\mu(t)\), such that
\[
W\mathcal ZW^*=\mathcal A .
\]
Moreover \(H(t)\ne0\) for almost every \(t\), and \(\mathcal H\) is separable. Explicitly, if \(a\) is a self-adjoint generator of \(\mathcal Z\), the construction gives \(WF(a)W^*=m_F\) for every bounded Borel function \(F\) on \(\mathbb R\).

**Proof.** *A generator and a measure.* By Fact 1.4(a), \(\mathcal Z\) is generated by a self-adjoint element \(a\). Let \((\xi_k)\) be a sequence that is dense in the unit ball of \(K\), and put
\[
\lambda(B)=\sum_k2^{-k}\mu_{\xi_k}(B)=\sum_k2^{-k}\|1_B(a)\xi_k\|^2
\]
for Borel sets \(B\subseteq\mathbb R\), with \(\mu_\xi\) as in Fact 1.4(b). This is a finite Borel measure, and \(\lambda(\mathbb R)=\sum_k2^{-k}\|\xi_k\|^2>0\), because some \(\xi_k\) is nonzero. Let \(\mu=\lambda/\lambda(\mathbb R)\), a probability measure with the same null sets as \(\lambda\).

*The calculus.* Put \(\Pi(F)=F(a)\) for bounded Borel functions \(F\) on \(\mathbb R\). Conditions (i) and (ii) of Fact 1.3 hold by Fact 1.4(b). For (iii), let \(F=0\) outside a Borel set \(N\) with \(\mu(N)=0\). The operator \(1_N(a)\) is a projection, by multiplicativity and \(\bar1_N=1_N\), and \(\|1_N(a)\xi_k\|^2=\mu_{\xi_k}(N)=0\) for every \(k\), because \(\lambda(N)=0\). So \(1_N(a)\) vanishes on a dense subset of the unit ball, hence \(1_N(a)=0\), and \(F(a)=(F1_N)(a)=F(a)1_N(a)=0\). For (iv), take \(\eta_k=\xi_k\) and \(F=1\). Fact 1.3, applied to \((\mathbb R,\text{Borel sets},\mu)\), gives Borel sets \(Z_k\), the measurable field \(H(t)=\overline{\operatorname{span}}\{\varepsilon_k:t\in Z_k\}\), and a unitary \(W\) with \(WF(a)W^*=m_F\) for every bounded Borel \(F\).

*The algebra.* Let \(\mathcal Z_0=\{F(a)\}\). Then \(W\mathcal Z_0W^*=\{m_F\}=\mathcal A\), which is a von Neumann algebra (Fact 1.1(f)); conjugation by a unitary carries commutants to commutants, so \(\mathcal Z_0\) is a von Neumann algebra. Each \(F(a)\) commutes with every operator that commutes with \(a\) (Fact 1.4(b)), so \(\mathcal Z_0\subseteq\{a\}''=\mathcal Z\). The continuous bounded function \(F(t)=\max(-\|a\|,\min(t,\|a\|))\) agrees with \(t\) on the spectrum of \(a\), so \(F(a)=a\) and \(a\in\mathcal Z_0\); hence \(\mathcal Z\subseteq\mathcal Z_0\). So \(\mathcal Z_0=\mathcal Z\) and \(W\mathcal ZW^*=\mathcal A\).

*Nonzero fibres.* Let \(\Gamma_0=\mathbb R\setminus\bigcup_kZ_k\), the Borel set where \(H(t)=0\). The function \(1_{\Gamma_0}\) vanishes on \(\{n\ge1\}\), so \(m_{1_{\Gamma_0}}=0\) (Fact 1.1(f)) and \(1_{\Gamma_0}(a)=W^*m_{1_{\Gamma_0}}W=0\). Hence \(\lambda(\Gamma_0)=\sum_k2^{-k}\|1_{\Gamma_0}(a)\xi_k\|^2=0\). Finally \(\mathcal H\) is separable by Fact 1.1(e), or simply because it is unitarily equivalent to \(K\). \(\square\)

**Remark 2.2.** Theorem 2.1 is the spectral theorem in multiplication form for one self-adjoint operator, read as a statement about the von Neumann algebra that the operator generates (Unbounded decomposable operators and measurable spectral decomposition, Remark 6.2). The fibre dimension \(n(t)\) is the number of \(k\) with \(t\in Z_k\). By Theorem 4.4(1), it does not depend on the choices, up to a Borel identification of the bases. It measures how often each part of the spectrum is repeated: \(\mathcal Z\) is maximal abelian exactly when \(n(t)\le1\) for almost every \(t\) (Example 6.2). If \(K=0\), the first statement of the theorem holds with \(H(t)=0\) for every \(t\) and any \(\mu\).

## 3. Lemmas on fibres

In this section and the next two, \(\Sigma\) is the Borel \(\sigma\)-algebra of a standard Borel space \(\Gamma\), and \(\mu\) is \(\sigma\)-finite. So \(\mathcal H\) is separable (Fact 1.1(e)), and every von Neumann algebra on \(\mathcal H\) is generated by a countable set (Fact 1.2(h)).

*Representing projections.* Let \(M\) be a measurable field and \(e\in\int^\oplus M\) a projection. By Fact 1.2(b), \(e=\int^\oplus x\) with \(x(\gamma)\in M(\gamma)\) for every \(\gamma\). The set \(N\) where \(x(\gamma)\ne x(\gamma)^*\) or \(x(\gamma)\ne x(\gamma)^2\) is measurable (Fact 1.1(d)), and it is null, since \(e=e^*=e^2\) (Fact 1.1(e)). The field \(1_{\Gamma\setminus N}x\) represents \(e\), and its values are projections in \(M(\gamma)\) at every point. We always represent projections of \(\int^\oplus M\) in this way and write \(e(\gamma)\) for the values.

**Lemma 3.1** (Dense fibre algebras). Let \(M\) be a measurable field. There are a countable set \(\mathcal Q\subseteq\int^\oplus M\), which is a unital \(*\)-algebra over the field \(\mathbb Q(i)=\mathbb Q+i\mathbb Q\), fields \(\beta_b\) with \(b=\int^\oplus\beta_b\) and \(\beta_b(\gamma)\in M(\gamma)\) for every \(\gamma\) (\(b\in\mathcal Q\)), and a null set \(N\), such that:

1. the complex linear span of \(\mathcal Q\) is dense in \(\int^\oplus M\) for the strong and for the \(\sigma\)-weak topology;
2. for \(\gamma\notin N\), the map \(b\mapsto\beta_b(\gamma)\) is a unital \(*\)-homomorphism of \(\mathbb Q(i)\)-algebras from \(\mathcal Q\) into \(M(\gamma)\), and the complex linear span of its image is dense in \(M(\gamma)\) for the strong and for the \(\sigma\)-weak topology.

**Proof.** By Fact 1.2(c) and (h), \(\int^\oplus M\) is a von Neumann algebra generated by a countable set \(S\). Let \(\mathcal Q\) be the set of noncommutative polynomials in the elements of \(S\cup S^*\) with coefficients in \(\mathbb Q(i)\), constants included. It is countable, it is a unital \(*\)-algebra over \(\mathbb Q(i)\), and it lies in \(\int^\oplus M\). Its complex linear span \(\mathcal Q_{\mathbb C}\) is a unital \(*\)-algebra containing \(S\cup S^*\), so its bicommutant is the von Neumann algebra generated by \(S\), which is \(\int^\oplus M\). Fact 1.5(b) gives (1). Choose the fields \(\beta_b\) by Fact 1.2(b).

For \(b,c\in\mathcal Q\) and \(q\in\mathbb Q(i)\), the operators \(b+c\), \(qb\), \(bc\) and \(b^*\) lie in \(\mathcal Q\), and they are represented both by \(\beta_{b+c},\beta_{qb},\beta_{bc},\beta_{b^*}\) and by \(\beta_b+\beta_c,q\beta_b,\beta_b\beta_c,\beta_b^*\) (Fact 1.1(e)). Two fields representing the same operator agree almost everywhere; likewise \(\beta_1=1\) almost everywhere. These are countably many identities, so they all hold off one null set \(N_1\). Off \(N_1\) the map \(b\mapsto\beta_b(\gamma)\) is a unital \(*\)-homomorphism of \(\mathbb Q(i)\)-algebras.

The von Neumann algebra generated by \(\mathcal A\cup S\) is \(\int^\oplus M\): it contains the algebra generated by \(S\), and it lies in the von Neumann algebra \(\int^\oplus M\), which contains \(\mathcal A\) (Fact 1.2(b)). By Fact 1.2(f), \(\int^\oplus M=\int^\oplus M_S\), where \(M_S(\gamma)\) is generated by \(\{\beta_s(\gamma):s\in S\}\). By Fact 1.2(d), \(M_S(\gamma)=M(\gamma)\) off a null set \(N_2\). Put \(N=N_1\cup N_2\) and let \(\gamma\notin N\). The complex span of \(\{\beta_b(\gamma):b\in\mathcal Q\}\) is a unital \(*\)-algebra contained in \(M(\gamma)\), and it contains every \(\beta_s(\gamma)\), \(s\in S\). So its bicommutant is \(M_S(\gamma)=M(\gamma)\), and Fact 1.5(b) gives the density in (2). \(\square\)

**Lemma 3.2** (Total families of sections). Let \(p\in\mathcal D\) be a projection, represented by a measurable field of projections \(p(\gamma)\) (as above, with Fact 1.1(f) in place of Fact 1.2(b)), and let \((\eta_j)\) be a sequence in \(p\mathcal H\). If the vectors \(m_f\eta_j\), with \(f\) bounded and measurable and \(j\ge1\), span a dense subspace of \(p\mathcal H\), then for almost every \(\gamma\) the vectors \(\eta_j(\gamma)\) span a dense subspace of \(p(\gamma)H(\gamma)\).

**Proof.** Since \(p\eta_j=\eta_j\), the sections \(p(\gamma)\eta_j(\gamma)\) and \(\eta_j(\gamma)\) agree off a null set \(N_0\) (Fact 1.1(e)); replace every \(\eta_j\) by \(1_{\Gamma\setminus N_0}\eta_j\), which changes no vector. Then \(\eta_j(\gamma)\in p(\gamma)H(\gamma)\) for every \(\gamma\) and \(j\). Let \(P(\gamma)\) be the projection onto the closed span \(L(\gamma)\) of \(\{\eta_j(\gamma)\}\). It is a measurable operator field (Fact 1.1(b)), and \(P(\gamma)\le p(\gamma)\). Put \(\hat P=\int^\oplus P\). Then \(\hat P\eta_j=\eta_j\), and \(\hat Pm_f\eta_j=m_f\hat P\eta_j=m_f\eta_j\) (Fact 1.1(e)). So \(\hat P\) is the identity on a dense subspace of \(p\mathcal H\), hence on \(p\mathcal H\): \(\hat Pp=p\). The fields \(Pp\) and \(p\) therefore agree almost everywhere, that is, \(P(\gamma)p(\gamma)=p(\gamma)\) for almost every \(\gamma\). For such \(\gamma\), \(p(\gamma)H(\gamma)\subseteq L(\gamma)\subseteq p(\gamma)H(\gamma)\). \(\square\)

**Lemma 3.3** (Fibres of a vector functional). Let \(\zeta=(\zeta_n)\) be a sequence in \(\mathcal H\) with \(\sum_n\|\zeta_n\|^2<\infty\). Then \(\sum_n\|\zeta_n(\gamma)\|^2<\infty\) off a null set \(N_\zeta\). Put \(\zeta(\gamma)=(\zeta_n(\gamma))_n\), and \(\zeta(\gamma)=0\) for \(\gamma\in N_\zeta\). For every essentially bounded measurable operator field \(x\), the function \(\gamma\mapsto\omega_{\zeta(\gamma)}(x(\gamma))\) is integrable, and
\[
\omega_\zeta\Big(\int^\oplus x\Big)=\int_\Gamma\omega_{\zeta(\gamma)}(x(\gamma))\,d\mu(\gamma).
\tag{3.1}
\]

**Proof.** By monotone convergence, \(\int\sum_n\|\zeta_n(\gamma)\|^2\,d\mu=\sum_n\|\zeta_n\|^2<\infty\), so the integrand is finite off a null set. Let \(\|x(\gamma)\|\le C\) off a null set. The functions \(\gamma\mapsto\langle x(\gamma)\zeta_n(\gamma),\zeta_n(\gamma)\rangle\) are measurable (Fact 1.1(a), (d)) and bounded in absolute value by \(C\|\zeta_n(\gamma)\|^2\) almost everywhere. These bounds have a finite total integral. So the series \(\sum_n\langle x(\gamma)\zeta_n(\gamma),\zeta_n(\gamma)\rangle\) converges absolutely almost everywhere, its sum is integrable, and its integral is the sum of the integrals, which is \(\sum_n\langle(\int^\oplus x)\zeta_n,\zeta_n\rangle\). \(\square\)

**Lemma 3.4** (Traces and faithfulness pass to the fibres). Let \(M\) be a measurable field, \(p\in\int^\oplus M\) a projection with values \(p(\gamma)\), and \(\zeta=(\zeta_n)\) a sequence in \(p\mathcal H\) with \(\sum_n\|\zeta_n\|^2<\infty\).

1. If \(\omega_\zeta\) is tracial on \(p(\int^\oplus M)p\), then for almost every \(\gamma\), \(\omega_{\zeta(\gamma)}\) is tracial on \(p(\gamma)M(\gamma)p(\gamma)\).
2. If \(\omega_\zeta\) is faithful on \(p(\int^\oplus M)p\), then for almost every \(\gamma\), the vectors \(y\zeta_n(\gamma)\), with \(y\in M(\gamma)'\) and \(n\ge1\), span a dense subspace of \(p(\gamma)H(\gamma)\); consequently \(\omega_{\zeta(\gamma)}\) is faithful on \(p(\gamma)M(\gamma)p(\gamma)\).

**Proof.** Take \(\mathcal Q\), \(\beta_b\) and \(N\) of Lemma 3.1 for \(M\).

(1) Let \(b,c\in\mathcal Q\) and let \(f\) be bounded and measurable. The operator \(m_f\) lies in the centre of \(\int^\oplus M\) (Fact 1.2(c)), so \(m_fb\in\int^\oplus M\), and by the hypothesis
\[
\omega_\zeta(m_f\,pbpcp)=\omega_\zeta(p(m_fb)pcp)=\omega_\zeta(pcp(m_fb)p)=\omega_\zeta(m_f\,pcpbp).
\]
Let \(g(\gamma)=\omega_{\zeta(\gamma)}\big(p\beta_bp\beta_cp(\gamma)-p\beta_cp\beta_bp(\gamma)\big)\), where \(p\beta_bp\beta_cp(\gamma)\) stands for \(p(\gamma)\beta_b(\gamma)p(\gamma)\beta_c(\gamma)p(\gamma)\). By (3.1), \(\int fg\,d\mu=0\). Taking \(f=\bar g/|g|\) on \(\{g\ne0\}\) and \(f=0\) elsewhere gives \(\int|g|\,d\mu=0\), so \(g=0\) off a null set \(N_{b,c}\). Let \(\gamma\) lie outside \(N\), \(N_\zeta\) and every \(N_{b,c}\), a countable union of null sets. Then \(\omega_{\zeta(\gamma)}(pypzp)=\omega_{\zeta(\gamma)}(pzpyp)\), with \(p=p(\gamma)\), for \(y,z\) in the image of \(\mathcal Q\), and by linearity for \(y,z\) in its complex span. For fixed \(z\) in that span, both sides are \(\sigma\)-weakly continuous functions of \(y\) (Fact 1.5(a); \(\omega_{\zeta(\gamma)}\) is \(\sigma\)-weakly continuous). By Lemma 3.1(2) the identity holds for every \(y\in M(\gamma)\). Then fix \(y\in M(\gamma)\) and let \(z\) vary in the same way.

(2) Lemma 3.1 also applies to the measurable field \(M'\) (Fact 1.2(a)), whose direct integral is \((\int^\oplus M)'\) (Fact 1.2(c)). It gives \(\mathcal Q'\subseteq(\int^\oplus M)'\), fields \(\beta'_c\) with values in \(M(\gamma)'\), and a null set \(N'\). Let \(r\) be the projection onto the closed span of \(\{c\zeta_n:c\in(\int^\oplus M)',\ n\ge1\}\). This subspace is invariant under the self-adjoint set \((\int^\oplus M)'\), so \(r\in(\int^\oplus M)''=\int^\oplus M\) (Fact 1.5(c)). It lies in \(p\mathcal H\), because \(c\zeta_n=cp\zeta_n=pc\zeta_n\); so \(r\le p\), and \(x=p-r\in\int^\oplus M\) satisfies \(x^*x=x\) and \(\omega_\zeta(px^*xp)=\sum_n\|(p-r)\zeta_n\|^2=0\). By faithfulness \(xp=p-r=0\). So the vectors \(c\zeta_n\), \(c\in(\int^\oplus M)'\), span a dense subspace of \(p\mathcal H\). By the strong density in Lemma 3.1(1), so do the vectors \(c\zeta_n\) with \(c\in\mathcal Q'\). These vectors are represented by the sections \(\beta'_c(\gamma)\zeta_n(\gamma)\), and Lemma 3.2 shows that, for almost every \(\gamma\), the vectors \(\beta'_c(\gamma)\zeta_n(\gamma)\) span a dense subspace of \(p(\gamma)H(\gamma)\). Since \(\beta'_c(\gamma)\in M(\gamma)'\), and since every vector \(y\zeta_n(\gamma)\) with \(y\in M(\gamma)'\) lies in \(p(\gamma)H(\gamma)\) for almost every \(\gamma\) (because \(p(\gamma)\in M(\gamma)\) commutes with \(y\), and \(p(\gamma)\zeta_n(\gamma)=\zeta_n(\gamma)\) almost everywhere), this proves the first claim.

For faithfulness, take such a \(\gamma\), outside \(N_\zeta\) and outside the null set where \(p(\gamma)\zeta_n(\gamma)\ne\zeta_n(\gamma)\) for some \(n\). Let \(x\in M(\gamma)\) with \(\omega_{\zeta(\gamma)}(p(\gamma)x^*xp(\gamma))=\sum_n\|xp(\gamma)\zeta_n(\gamma)\|^2=0\). Then \(x\zeta_n(\gamma)=0\) for all \(n\), so \(xy\zeta_n(\gamma)=yx\zeta_n(\gamma)=0\) for \(y\in M(\gamma)'\). Thus \(x\) vanishes on a dense subspace of \(p(\gamma)H(\gamma)\), and \(xp(\gamma)=0\). \(\square\)

**Lemma 3.5** (Finiteness from a trace). Let \(N\) be a von Neumann algebra on \(L\), \(p\in N\) a projection, and \(\zeta\) a square-summable sequence in \(L\) such that \(\omega_\zeta\) is tracial and faithful on \(pNp\). Then \(p\) is finite.

**Proof.** Let \(v\in N\) with \(v^*v=p\) and \(vv^*\le p\). Then \(v=vp=pv\). Traciality with \(x=v^*\), \(y=v\) gives \(\omega_\zeta(v^*v)=\omega_\zeta(pv^*pvp)=\omega_\zeta(pvpv^*p)=\omega_\zeta(vv^*)\). Put \(x=p-vv^*\), a projection below \(p\). Then \(\omega_\zeta(px^*xp)=\omega_\zeta(p)-\omega_\zeta(vv^*)=0\), and faithfulness gives \(x=xp=0\). So \(vv^*=p\). \(\square\)

**Corollary 3.6** (Disintegration of positive normal functionals). Let \(M\) be a measurable field and \(\varphi\) a \(\sigma\)-weakly continuous positive linear functional on \(\int^\oplus M\). For almost every \(\gamma\) there is a \(\sigma\)-weakly continuous positive linear functional \(\varphi_\gamma\) on \(M(\gamma)\), such that for every essentially bounded measurable operator field \(x\) with \(x(\gamma)\in M(\gamma)\) almost everywhere, the function \(\gamma\mapsto\varphi_\gamma(x(\gamma))\) is integrable and
\[
\varphi\Big(\int^\oplus x\Big)=\int_\Gamma\varphi_\gamma(x(\gamma))\,d\mu(\gamma).
\]
If \(\varphi\) is tracial, then \(\varphi_\gamma\) is tracial for almost every \(\gamma\); if \(\varphi\) is faithful, then \(\varphi_\gamma\) is faithful for almost every \(\gamma\).

**Proof.** By Fact 1.5(e), \(\varphi=\omega_\zeta\) on \(\int^\oplus M\) for a square-summable sequence \(\zeta\) in \(\mathcal H\). Put \(\varphi_\gamma=\omega_{\zeta(\gamma)}\) restricted to \(M(\gamma)\). Lemma 3.3 gives the formula, and Lemma 3.4 with \(p=1\) gives the last sentence. \(\square\)

## 4. The central decomposition

**Theorem 4.1** (Central decomposition). Let \(Q\) be a von Neumann algebra on a separable Hilbert space \(K\ne0\), with centre \(\mathcal Z\). Let \(\mu\), \((H(t))\) and \(W\) be as in Theorem 2.1 for \(\mathcal Z\), so that \(W\mathcal ZW^*=\mathcal A\).

1. There is a measurable field \(M\) of von Neumann algebras on \((H(t))\), with \(M(t)\) a factor for every \(t\), such that
\[
WQW^*=\int^\oplus M,\qquad WQ'W^*=\int^\oplus M' .
\]
If \(L\) is a measurable field with \(WQW^*=\int^\oplus L\), then \(L(t)=M(t)\) for almost every \(t\).
2. (*Generators.*) Let \(S\subseteq WQW^*\) be a countable set such that \(\mathcal A\cup S\) generates \(WQW^*\), for instance a set that generates \(WQW^*\) by itself, and let \(s=\int^\oplus\beta_s\) for \(s\in S\). Then \(M(t)\) is generated by \(\{\beta_s(t):s\in S\}\) for almost every \(t\). The same holds for \(WQ'W^*\) and the field \(M'\).

**Proof.** (1) Every element of \(Q\) commutes with \(\mathcal Z\subseteq Q'\), so \(Q\subseteq\mathcal Z'\). Hence \(\mathcal A=W\mathcal ZW^*\subseteq WQW^*\subseteq W\mathcal Z'W^*=\mathcal A'=\mathcal D\) (Fact 1.1(f)). By Fact 1.2(g), \(WQW^*=\int^\oplus M_0\) for a measurable field \(M_0\). The centre of \(WQW^*\) is \(W\mathcal ZW^*=\mathcal A\), so by Fact 1.2(e) there is a null set \(N\) off which \(M_0(t)\) is a factor. Let \(M(t)=M_0(t)\) for \(t\notin N\) and \(M(t)=\mathbb C1\) for \(t\in N\). Then \(M\) is equivalent to \(M_0\), so it is measurable with the same direct integral (Fact 1.2(b)), and \(M'\) is equivalent to \(M_0'\). By Fact 1.2(c), \(WQ'W^*=(WQW^*)'=\int^\oplus M_0'=\int^\oplus M'\). Uniqueness is Fact 1.2(d).

(2) The von Neumann algebra generated by \(\mathcal A\cup S\) is \(WQW^*=\int^\oplus M\), and by Fact 1.2(f) it is \(\int^\oplus M_S\) with \(M_S(t)\) generated by \(\{\beta_s(t)\}\). By Fact 1.2(d), \(M_S(t)=M(t)\) for almost every \(t\). For \(Q'\), note that \(\mathcal Z\subseteq Q'\), so \(\mathcal A\subseteq WQ'W^*=\int^\oplus M'\), and \(M'\) is measurable (Fact 1.2(a)); the same argument applies. \(\square\)

**Theorem 4.2** (Vectors and vector functionals). In the situation of Theorem 4.1, identify \(K\) with \(\mathcal H\) through \(W\) and \(Q\) with \(\int^\oplus M\).

1. If a sequence \((\zeta_n)\) in \(K\) is cyclic for \(Q\), then for almost every \(t\) the vectors \(\zeta_n(t)\) form a cyclic set for \(M(t)\). If \((\zeta_n)\) is separating for \(Q\), then for almost every \(t\) the \(\zeta_n(t)\) form a separating set for \(M(t)\).
2. Let \(\sum_n\|\zeta_n\|^2<\infty\). If \(\omega_\zeta(xy)=\omega_\zeta(yx)\) for all \(x,y\in Q\), then for almost every \(t\), \(\omega_{\zeta(t)}(xy)=\omega_{\zeta(t)}(yx)\) for all \(x,y\in M(t)\). If moreover \(\omega_\zeta\) is faithful on \(Q\), then \(\omega_{\zeta(t)}\) is faithful on \(M(t)\) for almost every \(t\).

In particular, a cyclic, or separating, vector \(\zeta\) of \(Q\) has fibres \(\zeta(t)\) that are cyclic, or separating, for almost every \(M(t)\); and if \(\omega_\zeta\) is a trace on \(Q\), then \(\omega_{\zeta(t)}\) is a trace on \(M(t)\) for almost every \(t\).

**Proof.** (1) Take \(\mathcal Q\), \(\beta_b\) and \(N\) of Lemma 3.1 for \(M\). The closed span of \(Q\{\zeta_n\}\) is \(K\), and by Lemma 3.1(1) it is also the closed span of the vectors \(b\zeta_n\), \(b\in\mathcal Q\). These are represented by the sections \(\beta_b(t)\zeta_n(t)\). Lemma 3.2, with \(p=1\), shows that for almost every \(t\) the vectors \(\beta_b(t)\zeta_n(t)\) span a dense subspace of \(H(t)\). For such \(t\notin N\), these vectors lie in \(M(t)\{\zeta_n(t)\}\), which is therefore cyclic. A separating set for \(Q\) is a cyclic set for \(Q'=\int^\oplus M'\) (Fact 1.5(c)), and the same argument for the measurable field \(M'\) shows that \(\{\zeta_n(t)\}\) is cyclic for \(M(t)'\) for almost every \(t\), hence separating for \(M(t)''=M(t)\).

(2) Apply Lemma 3.4 with \(p=1\). \(\square\)

### Uniqueness

The central decomposition depends on choices: the generator \(a\), the sequence \((\xi_k)\), the construction of Fact 1.3. Different choices give the same decomposition up to a change of base and a field of unitaries.

**Lemma 4.3** (Isomorphisms of diagonal algebras come from Borel isomorphisms). For \(i=1,2\), let \(\Gamma_i\) be a standard Borel space with a \(\sigma\)-finite measure \(\mu_i\), let \(\mathcal A_i\) be the diagonal algebra of a measurable field \((H_i(\gamma))\) over it, and put \(\Gamma_i^+=\{n_i\ge1\}\). Let \(\pi:\mathcal A_2\to\mathcal A_1\) be a \(*\)-isomorphism. There are Borel sets \(\Gamma_i'\subseteq\Gamma_i^+\) such that \(\Gamma_i^+\setminus\Gamma_i'\) is null, and a bijection \(\theta:\Gamma_1'\to\Gamma_2'\) such that \(\theta\) and \(\theta^{-1}\) are Borel, a Borel set \(B\subseteq\Gamma_2'\) is \(\mu_2\)-null exactly when \(\theta^{-1}(B)\) is \(\mu_1\)-null, and \(\pi(m_f)=m_{f\circ\theta}\) for every bounded Borel function \(f\) on \(\Gamma_2\), where \(f\circ\theta\) is put equal to \(0\) outside \(\Gamma_1'\).

**Proof.** *Projections.* By Fact 1.1(f), the projections of \(\mathcal A_i\) are the operators \(m_{1_B}\) with \(B\subseteq\Gamma_i^+\) Borel, and \(m_{1_B}=m_{1_{B'}}\) exactly when \(B\triangle B'\) is null. For Borel sets \(B_1,B_2,\ldots\subseteq\Gamma_i^+\), the projection \(m_{1_{\bigcup_kB_k}}\) is the least projection of \(\mathcal A_i\) that majorizes every \(m_{1_{B_k}}\): if \(m_{1_D}\) majorizes them, then \(m_{1_{B_k}}m_{1_D}=m_{1_{B_k}}\), so \(B_k\setminus D\) is null for every \(k\), and so is \(\bigcup_kB_k\setminus D\). A \(*\)-isomorphism maps projections to projections, preserves the order of projections (\(p\le q\) means \(pq=p\)) and the complement \(1-p\), and therefore preserves these least upper bounds.

*A point map.* The Borel set \(\Gamma_2^+\) is a standard Borel space, so it has Borel subsets \(B_1,B_2,\ldots\) that separate its points. Choose Borel sets \(C_k\subseteq\Gamma_1^+\) with \(\pi(m_{1_{B_k}})=m_{1_{C_k}}\). Let \(\sigma(y)=(1_{B_k}(y))_k\) for \(y\in\Gamma_2^+\) and \(\tau(x)=(1_{C_k}(x))_k\) for \(x\in\Gamma_1^+\), Borel maps into \(\{0,1\}^{\mathbb N}\). By Fact 1.8(b), \(\sigma(\Gamma_2^+)\) is Borel and \(\sigma\) is a Borel isomorphism onto it. The Borel sets \(R\subseteq\{0,1\}^{\mathbb N}\) with \(\pi(m_{1_{\sigma^{-1}(R)}})=m_{1_{\tau^{-1}(R)}}\) contain the sets \(\{c:c_k=1\}\), and they are closed under complements and countable unions, because preimages commute with these operations, \(m_{1_{\Gamma_i^+\setminus B}}=1-m_{1_B}\), and \(\pi\) preserves complements and countable least upper bounds. The sets \(\{c:c_k=1\}\) generate the Borel sets of \(\{0,1\}^{\mathbb N}\) (Fact 1.8(a)), so every Borel set \(R\) has this property. For \(R\) the complement of \(\sigma(\Gamma_2^+)\), it gives \(m_{1_{\tau^{-1}(R)}}=\pi(0)=0\), so \(D_1=\tau^{-1}(\sigma(\Gamma_2^+))\) has a null complement in \(\Gamma_1^+\). On \(D_1\) put \(\varphi=\sigma^{-1}\circ\tau\), a Borel map into \(\Gamma_2^+\). For Borel \(B\subseteq\Gamma_2^+\), the set \(R=\sigma(B)\) gives \(\pi(m_{1_B})=m_{1_{\varphi^{-1}(B)}}\). In particular \(\varphi^{-1}(B)\) is null when \(B\) is null. By linearity and uniform limits (a \(*\)-homomorphism is norm-decreasing), \(\pi(m_f)=m_{f\circ\varphi}\) for every bounded Borel \(f\). In the same way, \(\pi^{-1}\) gives a Borel set \(D_2\subseteq\Gamma_2^+\) with null complement in \(\Gamma_2^+\) and a Borel map \(\psi:D_2\to\Gamma_1^+\) with \(\pi^{-1}(m_{1_C})=m_{1_{\psi^{-1}(C)}}\) for Borel \(C\subseteq\Gamma_1^+\).

*Inverse maps.* For Borel \(C\subseteq\Gamma_1^+\), \(m_{1_C}=\pi(\pi^{-1}(m_{1_C}))=m_{1_{\varphi^{-1}(\psi^{-1}(C))}}\), so \(C\) and \((\psi\circ\varphi)^{-1}(C)\) differ by a null set. Apply this to Borel sets \(C'_1,C'_2,\ldots\) that separate the points of \(\Gamma_1^+\): outside a null set \(N_1\subseteq\Gamma_1^+\), which we take to contain \(\Gamma_1^+\setminus(D_1\cap\varphi^{-1}(D_2))\), the points \(\psi(\varphi(x))\) and \(x\) lie in the same sets \(C'_k\), so \(\psi(\varphi(x))=x\). In the same way \(\varphi(\psi(y))=y\) for \(y\in\Gamma_2^+\) outside a null set \(N_2\) that contains \(\Gamma_2^+\setminus(D_2\cap\psi^{-1}(D_1))\).

*The bijection.* Put \(\Gamma_1'=\Gamma_1^+\setminus(N_1\cup\varphi^{-1}(N_2))\) and \(\Gamma_2'=\Gamma_2^+\setminus(N_2\cup\psi^{-1}(N_1))\); their complements in \(\Gamma_i^+\) are null. For \(x\in\Gamma_1'\), \(\varphi(x)\notin N_2\) and \(\psi(\varphi(x))=x\notin N_1\), so \(\varphi(x)\in\Gamma_2'\); symmetrically \(\psi(\Gamma_2')\subseteq\Gamma_1'\). So \(\theta=\varphi|_{\Gamma_1'}\) is a bijection onto \(\Gamma_2'\) with inverse \(\psi|_{\Gamma_2'}\), and both are Borel. A Borel \(B\subseteq\Gamma_2'\) is null exactly when \(\theta^{-1}(B)\) is: one direction was shown for \(\varphi\), the other follows from \(B=\psi^{-1}(\theta^{-1}(B))\cap\Gamma_2'\) in the same way. Finally \(f\circ\varphi\) and \(f\circ\theta\) differ only on the null set \(D_1\setminus\Gamma_1'\) and outside \(\Gamma_1^+\), where diagonal operators do not see them. \(\square\)

**Theorem 4.4** (Uniqueness). For \(i=1,2\), let \(\Gamma_i\) be a standard Borel space with a \(\sigma\)-finite measure \(\mu_i\), \((H_i(\gamma))\) a measurable field over it with direct integral \(\mathcal H_i\), and \(W_i:K\to\mathcal H_i\) a unitary with \(W_i\mathcal ZW_i^*=\mathcal A_i\), for one abelian von Neumann algebra \(\mathcal Z\) on \(K\). Let \(\theta:\Gamma_1'\to\Gamma_2'\) be given by Lemma 4.3 for \(\pi(x)=W_1W_2^*xW_2W_1^*\). Then, after removing a null set from \(\Gamma_1'\) and its image from \(\Gamma_2'\):

1. there are a measurable \(\rho:\Gamma_1'\to(0,\infty)\) with \(\mu_2(\theta(B))=\int_B\rho\,d\mu_1\) for Borel \(B\subseteq\Gamma_1'\), and a measurable field of unitaries \(v(\gamma):H_2(\theta(\gamma))\to H_1(\gamma)\), \(\gamma\in\Gamma_1'\), such that for every \(\xi\in\mathcal H_2\)
\[
(W_1W_2^*\xi)(\gamma)=\rho(\gamma)^{1/2}\,v(\gamma)\,\xi(\theta(\gamma))\qquad\text{for almost every }\gamma\in\Gamma_1';
\tag{4.1}
\]
in particular \(\dim H_1(\gamma)=\dim H_2(\theta(\gamma))\);
2. if \(Q\) is a von Neumann algebra on \(K\) with \(\mathcal Z\subseteq Q\subseteq\mathcal Z'\), and \(W_iQW_i^*=\int^\oplus M_i\) for measurable fields \(M_i\), then \(M_1(\gamma)=v(\gamma)M_2(\theta(\gamma))v(\gamma)^*\) for almost every \(\gamma\in\Gamma_1'\).

So two central decompositions of \(Q\), for instance those obtained in Theorem 4.1 from two generators of \(\mathcal Z\), have unitarily equivalent fibres after a Borel identification of the bases.

**Proof.** Since \(\Gamma_i^+\setminus\Gamma_i'\) is null and \(H_i(\gamma)=0\) off \(\Gamma_i^+\), every vector of \(\mathcal H_i\) is represented by a section that vanishes off \(\Gamma_i'\), and we regard \(\mathcal H_i\) as the direct integral over \((\Gamma_i',\mu_i)\).

*The density.* The measure \(\nu(B)=\mu_2(\theta(B))\) on the Borel sets of \(\Gamma_1'\) is \(\sigma\)-finite and has the same null sets as \(\mu_1\) (Lemma 4.3). Fact 1.1(i) gives \(\rho\). For a Borel function \(g\ge0\) on \(\Gamma_2'\),
\[
\int_{\Gamma_1'}g(\theta(\gamma))\,\rho(\gamma)\,d\mu_1(\gamma)=\int_{\Gamma_1'}g\circ\theta\,d\nu=\int_{\Gamma_2'}g\,d\mu_2 :
\tag{4.2}
\]
for \(g=1_C\) both sides are \(\mu_2(C)\), and the general case follows by linearity and monotone convergence.

*Change of variables.* For \(\gamma\in\Gamma_1'\) put \(\tilde H(\gamma)=H_2(\theta(\gamma))\). If \((\xi_j)\) is a fundamental sequence of \((H_2(\gamma))\), the sections \(\xi_j\circ\theta\) have measurable Gram functions, compositions of measurable functions with the Borel map \(\theta\), and they are total in every \(\tilde H(\gamma)\). By Fact 1.1(g) they generate a measurable field \((\tilde H(\gamma))\), and \(\xi\circ\theta\) is a measurable section of it for every measurable section \(\xi\) of \((H_2(\gamma))\). Put \((R\xi)(\gamma)=\rho(\gamma)^{1/2}\xi(\theta(\gamma))\). By (4.2) with \(g=\|\xi\|^2\), \(R\) is an isometry of \(\mathcal H_2\) into \(\tilde{\mathcal H}=\int^\oplus_{\Gamma_1'}\tilde H\,d\mu_1\). It is onto: for a measurable section \(\eta\) of \((\tilde H(\gamma))\), the section \(\xi(y)=\rho(\theta^{-1}(y))^{-1/2}\eta(\theta^{-1}(y))\) of \((H_2(y))\) is measurable, by the same argument for \(\theta^{-1}\), and \(R\xi=\eta\). For an essentially bounded measurable operator field \(x\) on \((H_2(\gamma))\), \(R(\int^\oplus x)R^*=\int^\oplus x\circ\theta\), where \(x\circ\theta\) is a measurable operator field on \((\tilde H(\gamma))\) because it maps each \(\xi_j\circ\theta\) to the measurable section \((x\xi_j)\circ\theta\). In particular \(Rm_fR^*=m_{f\circ\theta}\).

*The unitary field.* Let \(U=W_1W_2^*R^*:\tilde{\mathcal H}\to\mathcal H_1\), a unitary. For bounded Borel \(f\) on \(\Gamma_2'\), Lemma 4.3 gives \(Um_{f\circ\theta}=W_1W_2^*m_fR^*=\pi(m_f)W_1W_2^*R^*=m_{f\circ\theta}U\). As \(\theta\) is a Borel isomorphism, every bounded Borel function on \(\Gamma_1'\) has the form \(f\circ\theta\). By Fact 1.1(h), \(U=\int^\oplus v\) for a measurable field of operators \(v(\gamma):\tilde H(\gamma)\to H_1(\gamma)\), unitary off a null set; remove that null set from \(\Gamma_1'\) and its image from \(\Gamma_2'\). Then \(W_1W_2^*=UR\), which is (4.1). A unitary between two Hilbert spaces preserves their dimension.

(2) By the previous step, \(R(\int^\oplus M_2)R^*=\int^\oplus(M_2\circ\theta)\), where \((M_2\circ\theta)(\gamma)=M_2(\theta(\gamma))\) is a measurable field: if measurable fields \(x_j\) generate \(M_2\) off a null set \(N\), the fields \(x_j\circ\theta\) generate \(M_2\circ\theta\) off the null set \(\theta^{-1}(N)\). By Fact 1.2(i), \(U(\int^\oplus M_2\circ\theta)U^*=\int^\oplus v(M_2\circ\theta)v^*\). Hence
\[
\int^\oplus M_1=W_1QW_1^*=UR\,(W_2QW_2^*)\,R^*U^*=\int^\oplus v(M_2\circ\theta)v^*,
\]
and Fact 1.2(d) gives \(M_1(\gamma)=v(\gamma)M_2(\theta(\gamma))v(\gamma)^*\) for almost every \(\gamma\). \(\square\)

## 5. The types of the fibres

In this section \(\Gamma\) is a standard Borel space with its Borel sets, \(\mu\) is \(\sigma\)-finite, \(M\) is a measurable field of von Neumann algebras on \((H(\gamma))\) such that \(M(\gamma)\) is a factor for almost every \(\gamma\), and \(Q=\int^\oplus M\). By Fact 1.2(e) the centre of \(Q\) is \(\mathcal A\). The central decomposition of Theorem 4.1 is an example. Every central projection of \(Q\) is \(m_{1_E}\) for a Borel set \(E\): if \(m_f\) is a projection, then \(f=\bar f=f^2\) almost everywhere on \(\{n\ge1\}\) (Fact 1.1(f)), so \(m_f=m_{1_E}\) with \(E=\{f=1\}\).

**Lemma 5.1** (Borel choice). Let \(X\) and \(Y\) be standard Borel spaces, \(\mu\) a \(\sigma\)-finite measure on \(X\), \(G\subseteq X\times Y\) a Borel set, and \(A\) its projection to \(X\). Then \(A\) is \(\mu\)-measurable, and there are a Borel set \(X'\subseteq A\), such that \(A\setminus X'\) is contained in a null set, and a Borel map \(s:X'\to Y\) with \((x,s(x))\in G\) for every \(x\in X'\).

**Proof.** Give \(X\) and \(Y\) Polish topologies that generate their Borel sets. Then \(G\) is a Souslin space (Fact 1.8(a)), and the projection \(\pi:G\to X\) is continuous. By Fact 1.8(c), \(A=\pi(G)\) is a Souslin set, hence \(\mu\)-measurable, and there is \(\varphi:A\to G\) with \(\pi\circ\varphi=\mathrm{id}_A\) such that \(\varphi^{-1}(B)\) is \(\mu\)-measurable for every Borel \(B\subseteq G\). Let \(\psi\) be the second coordinate of \(\varphi\); then \((x,\psi(x))\in G\) for \(x\in A\), and \(\psi^{-1}(C)\) is \(\mu\)-measurable for every Borel \(C\subseteq Y\).

Choose Borel sets \(C_1,C_2,\ldots\subseteq Y\) that separate the points of \(Y\), for instance the preimages of a countable base under a Borel isomorphism of \(Y\) with a Polish space. Choose Borel sets \(A_0\subseteq A\) and \(D_k\subseteq X\), and null sets \(N_0,N_k\), with \(A\setminus A_0\subseteq N_0\) and \(\psi^{-1}(C_k)\,\triangle\,D_k\subseteq N_k\); this is possible because these sets are \(\mu\)-measurable. The maps \(\sigma(y)=(1_{C_k}(y))_k\) and \(\delta(x)=(1_{D_k}(x))_k\), into \(\{0,1\}^{\mathbb N}\), are Borel, and \(\sigma\) is injective. By Fact 1.8(b), \(\sigma(Y)\) is Borel and \(\sigma\) is a Borel isomorphism onto it. On \(X'=A_0\setminus\bigcup_{k\ge0}N_k\), a Borel set, we have \(\delta=\sigma\circ\psi\). So \(s=\sigma^{-1}\circ\delta\) is a Borel map on \(X'\) that agrees with \(\psi\), and \(A\setminus X'\subseteq\bigcup_{k\ge0}N_k\). \(\square\)

**Proposition 5.2.** Let \(E\subseteq\Gamma\) be a Borel set and \(z=m_{1_E}\). Consider the central summand \(Qz\) on \(z\mathcal H\).

- (a) If \(Qz\) is finite, then \(M(\gamma)\) is finite for almost every \(\gamma\in E\).
- (b) If \(Qz\) is properly infinite, then \(M(\gamma)\) is properly infinite for almost every \(\gamma\in E\).
- (c) If \(Qz\) is of type I, then \(M(\gamma)\) is of type I for almost every \(\gamma\in E\).
- (d) If \(Qz\) is of type II, then \(M(\gamma)\) is of type II for almost every \(\gamma\in E\).
- (e) If \(Qz\) is of type III, then \(M(\gamma)\) is of type III for almost every \(\gamma\in E\).

**Proof.** We may discard from \(E\) the null set where \(M(\gamma)\) is not a factor, and we may ignore \(\gamma\) with \(H(\gamma)=0\), because the zero algebra has all five properties. Recall that projections and equivalence in \(Qz\) are those of \(Q\) below \(z\) (Fact 1.6(c)), and that the centre of \(Qz\) acts on the separable space \(z\mathcal H\), so it is \(\sigma\)-finite.

(a) By Fact 1.7(a), \(Qz\) has a faithful normal tracial state \(\tau\). By Fact 1.5(e), applied on \(z\mathcal H\), there is a square-summable sequence \(\zeta\) in \(z\mathcal H\) with \(\tau(xz)=\omega_\zeta(xz)=\omega_\zeta(zxz)\) for \(x\in Q\). Then \(\omega_\zeta\) is tracial and faithful on \(zQz=Qz\). By Lemma 3.4, for almost every \(\gamma\), \(\omega_{\zeta(\gamma)}\) is tracial and faithful on \(1_E(\gamma)M(\gamma)1_E(\gamma)\), which is \(M(\gamma)\) for \(\gamma\in E\). By Lemma 3.5, \(M(\gamma)\) is finite for almost every \(\gamma\in E\).

(b) By Fact 1.6(e) applied to \(Qz\), there are \(u,v\in Qz\) with \(u^*u=v^*v=z\) and \(uu^*+vv^*=z\). Represent them by fields \(u(\gamma),v(\gamma)\in M(\gamma)\) (Fact 1.2(b)). These identities between decomposable operators hold fibrewise almost everywhere (Fact 1.1(e)), and \(z(\gamma)=1\) on \(E\). So for almost every \(\gamma\in E\), \(u(\gamma)\) and \(v(\gamma)\) are isometries with \(u(\gamma)u(\gamma)^*+v(\gamma)v(\gamma)^*=1\). If \(H(\gamma)\ne0\), then \(v(\gamma)v(\gamma)^*\ne0\), so \(1\sim u(\gamma)u(\gamma)^*<1\) in \(M(\gamma)\), and \(1\) is infinite. A factor whose unit is infinite is properly infinite.

(c) By Fact 1.6(d) applied to \(Qz\), there is an abelian projection \(e\in Qz\) with central support \(z\). Take \(\mathcal Q\), \(\beta_b\) and \(N\) of Lemma 3.1 for \(M\). For \(b,c\in\mathcal Q\), the operators \(ebe\) and \(ece\) commute, so the fields \(e\beta_be\) and \(e\beta_ce\) commute off a null set \(N_{b,c}\). Off the union of \(N\) and the \(N_{b,c}\), the operators \(e(\gamma)ye(\gamma)\) and \(e(\gamma)y'e(\gamma)\) commute for \(y,y'\) in the image of \(\mathcal Q\), hence for \(y,y'\) in its complex span, and by strong density (Lemma 3.1(2) and Fact 1.5(a), one variable at a time) for all \(y,y'\in M(\gamma)\). So \(e(\gamma)\) is abelian in \(M(\gamma)\) for almost every \(\gamma\). Let \(F\) be the Borel set of \(\gamma\in E\) with \(e(\gamma)=0\) and \(n(\gamma)\ge1\) (Fact 1.1(c), (d)). Then \(m_{1_F}\) is a central projection below \(z\) with \(m_{1_F}e=0\), so \(e\le z-m_{1_F}\) and \(z=c(e)\le z-m_{1_F}\). Hence \(m_{1_F}=0\), and \(\mu(F)=0\) by Fact 1.1(f). So for almost every \(\gamma\in E\) with \(H(\gamma)\ne0\), the factor \(M(\gamma)\) has a nonzero abelian projection, and it is of type I.

(d) *A finite projection with a faithful trace.* The algebra \(Qz\) is semifinite, so by Fact 1.6(d) there is a finite projection \(f\in Qz\) with central support \(z\). As in (c), \(f(\gamma)\ne0\) for almost every \(\gamma\in E\) with \(H(\gamma)\ne0\). The algebra \(Q_f\) on \(f\mathcal H\) (Fact 1.5(d)) is finite (Fact 1.6(a)) and has a \(\sigma\)-finite centre, so it has a faithful normal tracial state (Fact 1.7(a)); by Fact 1.5(e) on \(f\mathcal H\) it is \(\omega_\zeta\) for a square-summable sequence \(\zeta\) in \(f\mathcal H\). Thus \(\omega_\zeta\) is tracial and faithful on \(fQf\). By Lemmas 3.4 and 3.5, for almost every \(\gamma\), \(\omega_{\zeta(\gamma)}\) is tracial and faithful on \(f(\gamma)M(\gamma)f(\gamma)\) and \(f(\gamma)\) is finite in \(M(\gamma)\).

*Division of \(f\).* An abelian projection of \(Q_f\) is the restriction of a projection \(e\le f\) of \(Q\) with \(eQe\) commutative; as \(Qz\) is of type II, \(e=0\). The algebra \(Q_f\) is finite, and below each of its nonzero central projections lies a nonzero finite projection, namely that central projection itself, which is a subprojection of the finite projection \(f\). So \(Q_f\) is of type II\(_1\). By Fact 1.7(b), for each \(k\ge1\) there are mutually orthogonal projections \(e_{k,1},\dots,e_{k,2^k}\le f\) in \(Q\) with sum \(f\), and partial isometries \(w_{k,j}\in fQf\) with \(w_{k,j}^*w_{k,j}=e_{k,1}\) and \(w_{k,j}w_{k,j}^*=e_{k,j}\). Represent all of them by fields with values in \(M(\gamma)\). Off a null set, these countably many relations hold in every fibre.

*No abelian projections in the fibres.* Fix \(\gamma\in E\) with \(H(\gamma)\ne0\) outside all the null sets above, and write \(\omega=\omega_{\zeta(\gamma)}\), \(f=f(\gamma)\), \(e_{k,j}=e_{k,j}(\gamma)\), and so on. Then \(w_{k,j}=e_{k,j}w_{k,j}e_{k,1}\in fM(\gamma)f\), so traciality gives \(\omega(e_{k,j})=\omega(w_{k,j}w_{k,j}^*)=\omega(w_{k,j}^*w_{k,j})=\omega(e_{k,1})\), and therefore \(\omega(e_{k,1})=2^{-k}\omega(f)\). Suppose \(q\) is a nonzero abelian projection of \(M(\gamma)\). Its only subprojections are \(0\) and \(q\) (Fact 1.6(a)). By Fact 1.6(b), either \(q\precsim e_{k,1}\) or \(e_{k,1}\prec q\). In the second case \(e_{k,1}\) is equivalent to a subprojection of \(q\) other than \(q\), that is, to \(0\); then every \(e_{k,j}\sim e_{k,1}\) is \(0\) and \(f=0\), which is excluded. So for every \(k\) there is a projection \(q_k\le e_{k,1}\) with \(q_k\sim q\). For each \(k\), a partial isometry \(v\in M(\gamma)\) with \(v^*v=q_1\) and \(vv^*=q_k\) satisfies \(v=q_kvq_1\in fM(\gamma)f\), so traciality gives \(\omega(q_1)=\omega(q_k)\le\omega(e_{k,1})=2^{-k}\omega(f)\). Hence \(\omega(q_1)=0\), and faithfulness gives \(q_1=0\), against \(q_1\sim q\ne0\). So \(0\) is the only abelian projection of \(M(\gamma)\), while \(f(\gamma)\) is a nonzero finite projection below its only nonzero central projection \(1\). So \(M(\gamma)\) is of type II.

(e) *The Borel set of finite projections.* Fix \(d\ge1\) and work on \(E_d=E\cap\Gamma_d\). By Fact 1.2(a), applied to \(M\) and to \(M'\), there are measurable operator fields \(x_k,y_k\) with norms at most \(1\) and a null set \(N_0\) such that for \(\gamma\notin N_0\) the \(x_k(\gamma)\) lie in \(M(\gamma)\) and are \(\sigma\)-weakly dense in its unit ball, and the \(y_k(\gamma)\) lie in \(M(\gamma)'\) and are \(\sigma\)-weakly dense in its unit ball. Transport them to the constant field: \(\tilde x_k(\gamma)=U(\gamma)x_k(\gamma)U(\gamma)^*\), \(\tilde y_k(\gamma)=U(\gamma)y_k(\gamma)U(\gamma)^*\) and \(\tilde M(\gamma)=U(\gamma)M(\gamma)U(\gamma)^*\), a von Neumann algebra on \(\ell^2_d\) with commutant \(U(\gamma)M(\gamma)'U(\gamma)^*\). The matrix entries of \(\tilde x_k\) and \(\tilde y_k\) are Borel functions on \(\Gamma_d\) (Fact 1.1(c), (d)).

Let \(\mathfrak B\) be the closed unit ball of \(B(\ell^2_d)\) with the \(\sigma\)-weak topology, a compact metrizable space (Fact 1.2(h)), and \(\mathfrak V\) the separable Hilbert space of square-summable sequences \(\zeta=(\zeta_n)\) in \(\ell^2_d\). Both are Polish. The functions \(p\mapsto\langle p\varepsilon_i,\varepsilon_j\rangle\) on \(\mathfrak B\) and \(\zeta\mapsto\langle\zeta_n,\varepsilon_i\rangle\) on \(\mathfrak V\) are continuous. Consider a finite product \(a\) of factors, each of which is the variable \(p\in\mathfrak B\) or one of the operators \(\tilde x_k(\gamma)\), \(\tilde y_k(\gamma)\). Its matrix entries are Borel functions of \((\gamma,p)\in E_d\times\mathfrak B\): this is clear for one factor, and for a product, \(\langle bc\,\varepsilon_i,\varepsilon_j\rangle=\sum_l\langle c\,\varepsilon_i,\varepsilon_l\rangle\langle b\,\varepsilon_l,\varepsilon_j\rangle\) is a pointwise convergent series of products of Borel functions. Consequently \(\langle a\zeta_n,\varepsilon_j\rangle=\sum_i\langle\zeta_n,\varepsilon_i\rangle\langle a\varepsilon_i,\varepsilon_j\rangle\), \(\langle a\zeta_n,\zeta_n\rangle=\sum_j\langle a\zeta_n,\varepsilon_j\rangle\overline{\langle\zeta_n,\varepsilon_j\rangle}\) and squared norms \(\|v\|^2=\sum_j|\langle v,\varepsilon_j\rangle|^2\) of finite linear combinations \(v\) of such vectors are Borel functions on \(E_d\times\mathfrak B\times\mathfrak V\) (Fact 1.8(a)). Let \(G_d\) be the set of \((\gamma,p,\zeta)\) with \(\gamma\in E_d\setminus N_0\) such that

- (G1) \(p=p^*=p^2\);
- (G2) \(p\tilde y_k(\gamma)=\tilde y_k(\gamma)p\) for every \(k\);
- (G3) \(\sum_n\|\zeta_n\|^2=1\) and \(p\zeta_n=\zeta_n\) for every \(n\);
- (G4) \(\sum_n\langle p\tilde x_k(\gamma)p\tilde x_l(\gamma)p\,\zeta_n,\zeta_n\rangle=\sum_n\langle p\tilde x_l(\gamma)p\tilde x_k(\gamma)p\,\zeta_n,\zeta_n\rangle\) for all \(k,l\);
- (G5) for every \(i\), the infimum of \(\|p\varepsilon_i-v\|\) over the finite linear combinations \(v\) of the vectors \(\tilde y_k(\gamma)\zeta_n\) with coefficients in \(\mathbb Q(i)\) is \(0\).

Each condition is a countable family of equalities between Borel functions, by the previous paragraph (for (G1), compare matrix entries of \(p\), \(p^*\) and \(p^2\); for (G2), those of \(p\tilde y_k\) and \(\tilde y_kp\); for (G3), the coordinates of \(p\zeta_n-\zeta_n\)). So \(G_d\) is a Borel subset of the standard Borel space \(\Gamma\times\mathfrak B\times\mathfrak V\).

*Points of \(G_d\) give finite projections with faithful traces.* Let \((\gamma,p,\zeta)\in G_d\). By (G2), \(p\) commutes with the \(\tilde y_k(\gamma)\), and by \(\sigma\)-weak density and Fact 1.5(a) with every element of the unit ball of \(\tilde M(\gamma)'\); so the projection \(p\) lies in \(\tilde M(\gamma)''=\tilde M(\gamma)\), and \(p\ne0\) by (G3). By (G4), \(\sigma\)-weak density and Fact 1.5(a), used one variable at a time, \(\omega_\zeta\) is tracial on \(p\tilde M(\gamma)p\). By (G5), every \(p\varepsilon_i\) lies in the closed span of the \(\tilde y_k(\gamma)\zeta_n\), which is contained in \(p\ell^2_d\) because \(\tilde y_k(\gamma)\zeta_n=\tilde y_k(\gamma)p\zeta_n=p\tilde y_k(\gamma)\zeta_n\); so the vectors \(\tilde y_k(\gamma)\zeta_n\) span a dense subspace of \(p\ell^2_d\). If \(x\in\tilde M(\gamma)\) and \(\omega_\zeta(px^*xp)=\sum_n\|x\zeta_n\|^2=0\), then \(x\tilde y_k(\gamma)\zeta_n=\tilde y_k(\gamma)x\zeta_n=0\), so \(xp=0\): \(\omega_\zeta\) is faithful on \(p\tilde M(\gamma)p\). By Lemma 3.5, \(p\) is a nonzero finite projection of \(\tilde M(\gamma)\). Transported back, \(U(\gamma)^*pU(\gamma)\) is a nonzero finite projection of \(M(\gamma)\), and the vectors \(U(\gamma)^*\zeta_n\) have the same properties with respect to it.

*Finite projections give points of \(G_d\).* Let \(\gamma\in E_d\setminus N_0\) and let \(q\) be a nonzero finite projection of \(M(\gamma)\). Put \(p=U(\gamma)qU(\gamma)^*\), so (G1) and (G2) hold. The algebra \(\tilde M(\gamma)_p\) on \(p\ell^2_d\) (Fact 1.5(d)) is finite (Fact 1.6(a)), and its centre is \(\sigma\)-finite, so it has a faithful normal tracial state, which is \(\omega_\zeta\) for a square-summable sequence \(\zeta\) in \(p\ell^2_d\) (Facts 1.7(a) and 1.5(e)). Then (G3) and (G4) hold. Let \(r\) be the projection onto the closed span \(L\) of \(\{y\zeta_n:y\in\tilde M(\gamma)',n\ge1\}\subseteq p\ell^2_d\). By Fact 1.5(d), the restrictions of \(\tilde M(\gamma)'\) to \(p\ell^2_d\) form the commutant of \(\tilde M(\gamma)_p\), and \(L\) is invariant under them, so the restriction of \(r\) to \(p\ell^2_d\) lies in \(\tilde M(\gamma)_p\) (Fact 1.5(c)). Since \(\omega_\zeta(p-r)=\sum_n\|(p-r)\zeta_n\|^2=0\), faithfulness gives \(r=p\), so \(L=p\ell^2_d\). For \(y\) in the unit ball of \(\tilde M(\gamma)'\), the vector \(y\zeta_n\) is a weak limit of vectors \(\tilde y_k(\gamma)\zeta_n\), by \(\sigma\)-weak density; the closed span of the \(\tilde y_k(\gamma)\zeta_n\) is weakly closed, so it contains \(L=p\ell^2_d\), and (G5) holds. So \((\gamma,p,\zeta)\in G_d\).

*Conclusion.* Let \(A_d\) be the projection of \(G_d\) to \(\Gamma\). Suppose \(A_d\) is not contained in a null set. Lemma 5.1 gives a Borel set \(X'\subseteq A_d\) and a Borel map \(s=(s_{\mathfrak B},s_{\mathfrak V})\) on \(X'\) with \((\gamma,s(\gamma))\in G_d\); since \(A_d\setminus X'\) lies in a null set, \(X'\) is not null, and we may shrink it to a Borel set with \(0<\mu(X')<\infty\). For \(\gamma\in X'\) put \(p(\gamma)=U(\gamma)^*s_{\mathfrak B}(\gamma)U(\gamma)\) and \(\zeta_n(\gamma)=U(\gamma)^*s_{\mathfrak V}(\gamma)_n\), and put \(p(\gamma)=0\) and \(\zeta_n(\gamma)=0\) for \(\gamma\notin X'\). The matrix entries of \(s_{\mathfrak B}\) and the coordinates of \(s_{\mathfrak V}(\cdot)_n\) are Borel functions on \(X'\), so \(p\) is a measurable operator field and the \(\zeta_n\) are measurable sections (Fact 1.1(a), (c), (d)). By the previous paragraphs, for \(\gamma\in X'\), \(p(\gamma)\) is a nonzero finite projection of \(M(\gamma)\), \(p(\gamma)\zeta_n(\gamma)=\zeta_n(\gamma)\), \(\sum_n\|\zeta_n(\gamma)\|^2=1\), and \(\omega_{\zeta(\gamma)}\) is tracial on \(p(\gamma)M(\gamma)p(\gamma)\) with the vectors \(M(\gamma)'\zeta_n(\gamma)\) dense in \(p(\gamma)H(\gamma)\). So \(P=\int^\oplus p\) is a projection of \(Q\), \(P\le z\) because \(X'\subseteq E\), and \(P\ne0\) because \(\|P\|=\operatorname*{ess\,sup}\|p(\gamma)\|=1\). The sequence \(\zeta=(\zeta_n)\) lies in \(P\mathcal H\), with \(\sum_n\|\zeta_n\|^2=\mu(X')<\infty\). For \(x,y\in Q\), represented by fields with values in \(M(\gamma)\), (3.1) gives
\[
\omega_\zeta(PxPyP)=\int_{X'}\omega_{\zeta(\gamma)}(pxpyp(\gamma))\,d\mu=\int_{X'}\omega_{\zeta(\gamma)}(pypxp(\gamma))\,d\mu=\omega_\zeta(PyPxP),
\]
in the notation of the proof of Lemma 3.4. If \(\omega_\zeta(Px^*xP)=0\), then \(\sum_n\|x(\gamma)\zeta_n(\gamma)\|^2=0\) for almost every \(\gamma\in X'\); for such \(\gamma\), \(x(\gamma)\) vanishes on the dense subspace spanned by \(M(\gamma)'\{\zeta_n(\gamma)\}\) of \(p(\gamma)H(\gamma)\), so \(x(\gamma)p(\gamma)=0\); and \(p(\gamma)=0\) off \(X'\). So \(xP=0\). By Lemma 3.5, \(P\) is a nonzero finite projection of \(Q\) below \(z\), which contradicts the hypothesis that \(Qz\) is purely infinite. Hence \(A_d\) is contained in a null set.

By the paragraph *Finite projections give points of \(G_d\)*, for \(\gamma\in E_d\setminus(N_0\cup A_d)\) the algebra \(M(\gamma)\) has no nonzero finite projection. As this holds for every \(d\ge1\), and \(H(\gamma)=0\) on \(\Gamma_0\), \(M(\gamma)\) is purely infinite for almost every \(\gamma\in E\). \(\square\)

**Theorem 5.3** (Types of the fibres). Let \(P\) be one of the properties: finite, properly infinite, semifinite, of type I, of type II, of type II\(_1\), of type II\(_\infty\), of type III. Then \(Q\) has \(P\) if and only if \(M(\gamma)\) has \(P\) for almost every \(\gamma\). More precisely, let \(z_T=m_{1_{E_T}}\) be the projections of the type decomposition of \(Q\) (Fact 1.6(c)), with Borel sets \(E_T\). Then for almost every \(\gamma\) with \(H(\gamma)\ne0\), \(\gamma\) lies in exactly one \(E_T\), and \(M(\gamma)\) is of type \(T\).

**Proof.** *Fibres over the summands.* The projections \(z_T\) are orthogonal with sum \(1\), so \(\sum_T1_{E_T}=1\) almost everywhere on \(\{n\ge1\}\) (Fact 1.1(f)); this is the first claim. Apply Proposition 5.2 to the summands \(Qz_T\). For almost every \(\gamma\in E_{\rm I}\), \(M(\gamma)\) is of type I, by (c). For almost every \(\gamma\in E_{{\rm II}_1}\), it is of type II and finite, by (d) and (a), so of type II\(_1\). The algebra \(Qz_{{\rm II}_\infty}\) is properly infinite, because \(0\) is its only finite central projection; so for almost every \(\gamma\in E_{{\rm II}_\infty}\), \(M(\gamma)\) is of type II and properly infinite, by (d) and (b), so of type II\(_\infty\). For almost every \(\gamma\in E_{\rm III}\), it is of type III, by (e).

*Types I, II\(_1\), II\(_\infty\), III.* Fix one of these types \(T\). If \(Q\) is of type \(T\), the uniqueness in Fact 1.6(c) gives \(z_T=1\), so almost every \(\gamma\) with \(H(\gamma)\ne0\) lies in \(E_T\), and \(M(\gamma)\) is of type \(T\). Conversely, let almost every \(M(\gamma)\) be of type \(T\), and let \(T'\ne T\). For almost every \(\gamma\in E_{T'}\) with \(H(\gamma)\ne0\), the nonzero factor \(M(\gamma)\) would be of the two types \(T\) and \(T'\). So \(E_{T'}\cap\{n\ge1\}\) is contained in a null set, and \(z_{T'}=0\). Hence \(z_T=1\), and \(Q\) is of type \(T\).

*Type II and semifiniteness.* A nonzero factor is of type II exactly when it is of type II\(_1\) or II\(_\infty\), and it is semifinite exactly when it is not of type III. On the side of \(Q\): \(Q\) is of type II exactly when \(z_{\rm I}=z_{\rm III}=0\). Indeed, if \(Q\) is of type II, its central summands \(Qz_{\rm I}\) and \(Qz_{\rm III}\) are of type II, and a nonzero algebra of type II is neither of type I (it has no nonzero abelian projection) nor of type III (it has a nonzero finite projection). Conversely, if \(z_{\rm I}=z_{\rm III}=0\), every abelian projection \(e\) of \(Q\) has \(ez_{{\rm II}_1}=ez_{{\rm II}_\infty}=0\), and every nonzero central projection \(c\) has \(cz_{{\rm II}_1}\ne0\) or \(cz_{{\rm II}_\infty}\ne0\), so it majorizes a nonzero finite projection. In the same way, \(Q\) is semifinite exactly when \(z_{\rm III}=0\). If \(z_{\rm III}\ne0\), then \(Qz_{\rm III}\) is a nonzero central summand of type III. If \(Qz\) is a nonzero central summand of type III, then \(Qzz_T\) is a central summand both of an algebra of type III and of one of type \(T\), so it is \(0\) for \(T\ne{\rm III}\) by the same comparison of types; hence \(z\le z_{\rm III}\) and \(z_{\rm III}\ne0\).

Now let \(Q\) be of type II. Then \(z_{\rm I}=z_{\rm III}=0\), so almost every \(\gamma\) with \(H(\gamma)\ne0\) lies in \(E_{{\rm II}_1}\cup E_{{\rm II}_\infty}\), and \(M(\gamma)\) is of type II. Conversely, let almost every \(M(\gamma)\) be of type II. For almost every \(\gamma\in E_{\rm I}\cup E_{\rm III}\) with \(H(\gamma)\ne0\), the nonzero factor \(M(\gamma)\) would be of type II and of type I or III; so \((E_{\rm I}\cup E_{\rm III})\cap\{n\ge1\}\) is contained in a null set, \(z_{\rm I}=z_{\rm III}=0\), and \(Q\) is of type II. The same argument with \(E_{\rm III}\) alone shows that \(Q\) is semifinite exactly when almost every fibre is semifinite.

*Finite and properly infinite.* Write \(1=e_{\rm f}+e_{\rm pi}\) as in Fact 1.6(c). These projections are central: \(c(e_{\rm f})e_{\rm pi}=c(e_{\rm f})c(e_{\rm pi})e_{\rm pi}=0\), so \(c(e_{\rm f})=c(e_{\rm f})(e_{\rm f}+e_{\rm pi})=e_{\rm f}\), and \(e_{\rm pi}=1-e_{\rm f}\). Let \(e_{\rm f}=m_{1_{E_{\rm f}}}\) and \(e_{\rm pi}=m_{1_{E_{\rm pi}}}\). By Proposition 5.2(a) and (b), almost every fibre over \(E_{\rm f}\) is finite and almost every fibre over \(E_{\rm pi}\) is properly infinite; a nonzero factor is not both. If \(Q\) is finite, then \(e_{\rm pi}\), a properly infinite projection below the finite projection \(1\), is finite, so \(e_{\rm pi}=0\); if \(Q\) is properly infinite, the finite central projection \(e_{\rm f}\) is \(0\). Conversely, if almost every fibre is finite, then almost every fibre over \(E_{\rm pi}\) is zero, so \(e_{\rm pi}=0\) and \(Q=Qe_{\rm f}\) is finite; if almost every fibre is properly infinite, then \(e_{\rm f}=0\) and \(Q\) is properly infinite. \(\square\)

**Remark 5.4.** The first step of the proof shows that the set of \(\gamma\) for which \(M(\gamma)\) is of a given type is measurable up to a null set: it agrees with \(E_T\cup\{n=0\}\) outside a null set. No Borel structure on the class of factors of a given type is needed for this. Measurable selection enters only in Proposition 5.2(e), to produce a finite projection of \(Q\) out of finite projections of the fibres.

## 6. Examples

**Example 6.1** (Factors). Let \(Q\) be a factor on a separable \(K\ne0\). Then \(\mathcal Z=\mathbb C1\), its self-adjoint generator \(a\) is a real multiple \(c1\) of the identity, every \(\mu_\xi\) is \(\|\xi\|^2\) times the point mass at \(c\), and \(\mu=\delta_c\). The direct integral over \(\delta_c\) is the single fibre \(H(c)\), and Theorem 4.1 is the identity \(Q\cong M(c)\). Exercise 7.1 gives the converse.

**Example 6.2** (Abelian algebras). Let \(Q=\mathcal Z\) be abelian. Then \(WQW^*=\mathcal A=\int^\oplus\mathbb C1\), so by uniqueness \(M(t)=\mathbb C1_{H(t)}\) for almost every \(t\), and \(WQ'W^*=\int^\oplus B(H(t))\,d\mu(t)=\mathcal D\). Every fibre is a factor of type I, and so \(Q\) is of type I, in line with Theorem 5.3. The algebra \(Q\) is maximal abelian, \(Q'=Q\), exactly when \(\mathcal D=\mathcal A\), that is, when \(B(H(t))=\mathbb C1\) for almost every \(t\) (Direct integrals of von Neumann algebras, Corollary 4.2), that is, when \(n(t)\le1\) almost everywhere.

**Example 6.3** (Constant fibres). Let \((\Gamma,\mu)\) be a standard Borel space with a \(\sigma\)-finite measure, \(K_0\ne0\) a separable Hilbert space and \(M_0\) a factor on \(K_0\). By Direct integrals of von Neumann algebras, Example 7.2, the algebra \(L^\infty(\Gamma,\mu)\mathbin{\bar\otimes}M_0\) on \(L^2(\Gamma,\mu)\otimes K_0\) is unitarily equivalent to the direct integral of the constant field \(M(\gamma)=M_0\), and its centre corresponds to \(\mathcal A\). So Theorem 5.3 applies: if \(\mu\ne0\), then \(L^\infty(\Gamma,\mu)\mathbin{\bar\otimes}M_0\) is finite, properly infinite, or of type I, II\(_1\), II\(_\infty\) or III exactly when \(M_0\) is.

**Example 6.4** (Atoms). Let \(Q=Q_1\oplus Q_2\) on \(K_1\oplus K_2\), with factors \(Q_1\ne0\) and \(Q_2\ne0\) on separable spaces. The centre is \(\mathbb C1\oplus\mathbb C1\) (The double commutant theorem, Proposition 5.2), generated by \(a=0\oplus1\). Then \(\mu\) is carried by the two points \(0\) and \(1\), with \(\mu(\{0\})>0\) and \(\mu(\{1\})>0\), and the fibres over \(0\) and \(1\) are unitarily equivalent to \(K_1\) and \(K_2\). Indeed \(W\) carries \(K_1=1_{\{0\}}(a)K\) onto \(m_{1_{\{0\}}}\mathcal H\), the sections supported in \(\{0\}\), which evaluation at \(0\) identifies with \(H(0)\); an operator \(\int^\oplus x\) acts there as \(x(0)\), and every element of \(M(0)\) occurs, as the value at \(0\) of a field that vanishes elsewhere. So \(M(0)\) is the image of \(Q_1\) under this identification, and likewise \(M(1)\) is that of \(Q_2\). If \(Q_1\) is finite and \(Q_2\) properly infinite, then \(Q\) is neither finite nor properly infinite, and accordingly neither property holds for almost every fibre.

## 7. Exercises

**Exercise 7.1.** In Theorem 2.1, show that \(\mathcal Z=\mathbb C1\) if and only if \(\mu\) is a point mass.

*Solution.* If \(\mathcal Z=\mathbb C1\), Example 6.1 shows \(\mu=\delta_c\). Conversely, let \(\mu=\delta_c\). Every bounded Borel \(F\) equals the constant \(F(c)\) almost everywhere, so \(m_F=F(c)1\), and \(\mathcal A=\mathbb C1\). Then \(\mathcal Z=W^*\mathcal AW=\mathbb C1\).

**Exercise 7.2.** Let \(Q\) be a von Neumann algebra on a separable \(K\ne0\) with a vector \(\xi\) that is cyclic and separating and whose vector functional \(\omega_\xi\) is a trace on \(Q\). In the central decomposition, show that for almost every \(t\) with \(H(t)\ne0\) the vector \(\xi(t)\) is nonzero, cyclic and separating for \(M(t)\), and that \(\omega_{\xi(t)}\) is a trace on \(M(t)\).

*Solution.* Theorem 4.2 gives cyclicity, separation and traciality for almost every \(t\). If \(H(t)\ne0\) and \(\xi(t)\) is cyclic, then \(\xi(t)\ne0\), because \(M(t)\{0\}=\{0\}\) is not dense in \(H(t)\).

**Exercise 7.3.** Let \(Q\) be of type II\(_1\) on a separable space, with central decomposition \(\int^\oplus M\). Show directly from Theorem 4.2 and Fact 1.7(a) that almost every \(M(t)\) has a faithful normal tracial positive functional, and deduce that almost every \(M(t)\) is finite.

*Solution.* By Fact 1.7(a) and Fact 1.5(e), \(Q\) has a faithful normal tracial state \(\omega_\zeta\). By Theorem 4.2(2), \(\omega_{\zeta(t)}\) is tracial and faithful on \(M(t)\) for almost every \(t\). Lemma 3.5 with \(p=1\) shows that \(M(t)\) is finite.

**Exercise 7.4.** Show that in Theorem 5.3 the words "almost every" cannot be replaced by "every": give a direct integral of factors that is finite although one of its fibres is properly infinite. (Use a field that differs from \(\mathbb C1\) at a single point of a base without atoms.)

*Solution.* Let \(\Gamma=[0,1]\) with Lebesgue measure on its Borel sets, and let \(H(\gamma)=K_0\) be the constant field of a separable infinite-dimensional space \(K_0\). Let \(M(\gamma)=\mathbb C1\) for \(\gamma\ne0\) and \(M(0)=B(K_0)\). Every \(M(\gamma)\) is a factor. The field agrees with \(\mathbb C1\) off the null set \(\{0\}\), so it is measurable and \(\int^\oplus M=\int^\oplus\mathbb C1=\mathcal A\) (Fact 1.2(b) and Direct integrals of von Neumann algebras, Proposition 2.2(5)). This algebra is abelian, so its unit is an abelian, hence finite, projection (Fact 1.6(a)). But \(B(K_0)\) is properly infinite: an isometry of \(K_0\) onto a proper closed subspace, which exists because \(K_0\) has an infinite orthonormal basis, shows \(1\sim f<1\).

## Where this leads

The classification of injective factors writes an injective algebra of type II\(_1\) as \(Z(Q)\mathbin{\bar\otimes}R\). It takes the central decomposition, uses Theorem 5.3 and Exercise 7.2 to see that almost every fibre is a factor of type II\(_1\) in standard form, and chooses isomorphisms of the fibres with \(R\) in a measurable way. The same decomposition reduces statements about injectivity, or about the representations of a group, to factors. Corollary 3.6 disintegrates positive normal functionals along any direct integral of von Neumann algebras over a standard base.

## References

- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C\*-Algebras and von Neumann Algebras*, revised and corrected edition with hyperlinks, free from the author: https://bruceblackadar.com/Mathematics/Cycr.pdf (first published as Encyclopaedia of Mathematical Sciences 122, 2006; the numbering is the same).