Central decomposition of a von Neumann algebra

Definition

Let $M\subseteq B(H)$ be a von Neumann algebra on a separable Hilbert space. A central decomposition of $M$ is a spatial direct-integral realization

H\cong\int_X^\oplus H_x\,d\mu(x),\qquad M\cong\int_X^\oplus M_x\,d\mu(x),

over a standard measure space, such that $M_x\subseteq B(H_x)$ is a factor for almost every $x$ , and the center $Z(M)$ corresponds to the diagonal scalar algebra $L^\infty(X,\mu)$ . Equalities and fiber properties are understood modulo null sets; individual fibers are not pointwise canonical.

Existence and uniqueness

For a separably acting von Neumann algebra, the factorial decomposition theorem produces such a measurable field of factors. Subject to the standard measurability hypotheses, the base measure class and factor field are unique up to the appropriate almost-everywhere measurable equivalence. This is a disintegration theorem, not merely a decomposition by finitely many central projections Kadison–Ringrose, vol. II, §6.5.

How the center controls the fibers

Under the decomposition, a central element $z$ acts on $H_x$ as multiplication by a scalar $f_z(x)$ . Conversely, every essentially bounded measurable scalar field gives a central decomposable operator. The fiber algebras have trivial centers almost everywhere, so the variation that remains in $Z(M)$ has been transferred to the parameter space Takesaki, Chapter IV, §8.

Examples and scope

A finite direct sum $M_1\oplus\cdots\oplus M_n$ of factors is the discrete case, with $X=\{1,\ldots,n\}$ . A commutative von Neumann algebra decomposes into one-dimensional factors almost everywhere. Separability or standardness cannot simply be dropped: more general von Neumann algebras require additional measure-theoretic formulations, and “the” decomposition should not be read as a preferred choice of representatives at every point.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, American Mathematical Society, 1997. AMS DOI record. Relevant: §6.5 on central and factorial decomposition.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. Springer DOI record. Relevant: Chapter IV, §8 on direct integrals and decomposable operators.