nLab measurable field of Hilbert spaces

Idea

A measurable field of Hilbert spaces is the exact analogue of a vector bundle over a topological space in the setting of fiber bundles of infinite-dimensional Hilbert spaces over measurable spaces.

Definition

The original definition is due to John von Neumann (Definition 1 in Neumann).

We present here a slightly modernized version, which can be found in many modern sources, e.g., Takesaki.

Definition

Suppose $(X,\Sigma)$ is a measurable space equipped with a σ-finite measure? $\mu$, or, less specifically, with a σ-ideal $N$ of negligible subsets so that $(X,\Sigma,N)$ is an enhanced measurable space. A measurable field of Hilbert spaces over $(X,\Sigma,N)$ is a family $H_x$ of Hilbert spaces indexed by points $x\in X$ together with a vector subspace? $M$ of the product $P$ of vector spaces $\prod_{x\in X} H_x$. The elements of $M$ are known as measurable sections. The pair $(\{H_x\}_{x\in X},M)$ must satisfy the following conditions.

• For any $m\in M$ the function $X\to\mathbf{R}$ ($x\mapsto \|m(x)\|$) is a measurable function on $(X,\Sigma)$.
• If for some $p\in P$, the function $X\to\mathbf{C}$ ($x\mapsto\langle p(x),m(x)\rangle$) is a measurable function on $(X,\Sigma)$ for any $m\in M$, then $p\in M$.
• There is a countable subset $M'\subset M$ such that for any $x\in X$, the closure of the span of vectors $m(x)$ ($m\in M'$) coincides with $H_x$.

The last condition restrict us to bundles of separable Hilbert spaces. One can also define bundles of nonseparable Hilbert spaces, but this cannot be done simply by dropping the last condition.

Serre–Swan-type duality

The category of measurable fields of Hilbert spaces on $(X,\Sigma,N)$ (as defined above) is equivalent to the category of countably-generated W*-modules over the commutative von Neumann algebra $\mathrm{L}^\infty(X,\Sigma,N)$.

(If we work with bundles of general, possibly nonseparable Hilbert spaces, then the W*-modules do not need to be countably generated.)

Related entries

• duality between algebra and geometry

• Serre-Swan theorem

• W*-module

• commutative von Neumann algebra

References

• John von Neumann, On Rings of Operators. Reduction Theory, The Annals of Mathematics 50:2 (1949), 401. doi.

• Masamichi Takesaki, Theory of Operator Algebras. I, Springer, 1979.