nLab
measurable field of Hilbert spaces

Contents

Contents

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 Takesaki.

Definition

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

  • For any mMm\in M the function XRX\to\mathbf{R} (xm(x)x\mapsto \|m(x)\|) is a measurable function on (X,Σ)(X,\Sigma).
  • If for some pPp\in P, the function XCX\to\mathbf{C} (xp(x),m(x)x\mapsto\langle p(x),m(x)\rangle) is a measurable function on (X,Σ)(X,\Sigma) for any mMm\in M, then pMp\in M.
  • There is a countable subset MMM'\subset M such that for any xXx\in X, the closure of the span of vectors m(x)m(x) (mMm\in M') coincides with H xH_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,Σ,N)(X,\Sigma,N) (as defined above) is equivalent to the category of countably-generated W*-modules over the commutative von Neumann algebra L (X,Σ,N)\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.)

References

Last revised on October 19, 2021 at 01:01:07. See the history of this page for a list of all contributions to it.