Knowlpedia

Construction

Direct integral of Hilbert spaces

The Hilbert space of square-integrable measurable sections of a measurable field of Hilbert spaces.

Core idea

Let (X,Σ,μ)(X,\Sigma,\mu) be a and {Hx}x∈X\{\mathcal H_x\}_{x\in X} a . Its direct integral is

∫X⊕Hx dμ(x),\int_X^\oplus \mathcal H_x\,d\mu(x),

the space of measurable sections ξ\xi, modulo equality , for which the ∫X∥ξ(x)∥2 dμ(x)\int_X\|\xi(x)\|^2\,d\mu(x) is finite. Addition and scalar multiplication are fiberwise, and

⟨ξ,η⟩=∫X⟨ξ(x),η(x)⟩Hx dμ(x).\langle\xi,\eta\rangle=\int_X\langle\xi(x),\eta(x)\rangle_{\mathcal H_x}\,d\mu(x).

After quotienting by null sections, this is a . The construction is the measure-theoretic analogue of a Hilbert direct sum.

Constant fields and direct sums

For a constant field Hx=K\mathcal H_x=K, the direct integral is the vector-valued space L2(X,μ;K)L^2(X,\mu;K). If XX has counting measure, it reduces to the Hilbert direct sum ⨁x∈XHx\bigoplus_{x\in X}\mathcal H_x. Atomic and continuous parts of the measure therefore interpolate between discrete sums and continuously indexed decompositions.

Decomposable operators

Suppose Tx:Hx→KxT_x:\mathcal H_x\to\mathcal K_x is a measurable field of bounded operators with essentially bounded norms. Then

(∫X⊕Tx dμ(x))ξ(x)=Txξ(x)\left(\int_X^\oplus T_x\,d\mu(x)\right)\xi(x)=T_x\xi(x)

defines a bounded operator between the corresponding direct integrals. Such preserve the fiberwise structure and are basic tools in spectral and representation-theoretic decompositions.

Conventions and use

The phrase “measurable field” includes the chosen measurable structure on sections; a bare family of Hilbert spaces is not enough. Changing fibers or sections on a does not change the direct integral. Standard decomposition theorems usually impose σ\sigma-finiteness and separability hypotheses, which must be stated separately from the construction Takesaki, Chapter IV, §8.

References
  1. Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV, §8 on direct integrals.
  2. Jacques Dixmier, Von Neumann Algebras, North-Holland, 1981. Publisher record. Relevant: Chapter II, §1 on measurable fields and direct integrals.