Direct integrals of unitary representations

Assume the Axiom of Choice (The Axiom of Choice). Let $G$ be a locally compact Hausdorff group, let $(X,\mathcal{B},\mu )$ be a sigma-finite standard-Borel measure space (Standard Borel spaces, Finite, sigma-finite, and semifinite measures), and let $(H_x,e_n(x){)}_{x\in X}$ be a measurable complex Hilbert field with a countable fundamental family (Measurable Hilbert field from a countable fundamental family). Write $\mathcal{H}={\int }_X^{\oplus }{H}_{x}d\mu (x)$ for its direct-integral Hilbert space (Direct integral of a measurable Hilbert field, Direct integrals of measurable Hilbert fields are Hilbert spaces). A measurable field of unitary representations of $G$ over this field is a family $({\pi }_x{)}_{x\in X}$ such that each ${\pi }_x:G\to U(H_x)$ is strongly continuous (Strongly continuous unitary representations, invariant linear subspaces and intertwiners) and, for each fixed $g\in G$, the operator field $x\mapsto {\pi }_x(g)$ is weakly measurable (Measurable and decomposable operator fields). The identities ${\pi }_x(gh)={\pi }_x(g){\pi }_x(h)$ and ${\pi }_x(e)=I_{H_x}$ are required for every $x\in X$ and all $g,h\in G$. For each $g\in G$ define $\pi (g)={\int }_X^{\oplus }{\pi }_x(g)d\mu (x)\in \mathcal{B}(\mathcal{H})\mathrm{.}$ By Measurable essentially bounded operator fields act decomposably, these operators define a group homomorphism $G\to U(\mathcal{H})$. This operator family is called the direct integral of the field and written $\pi ={\int }_X^{\oplus }{\pi }_{x}d\mu (x)$. The homomorphism and unitarity are proved below; strong continuity is not asserted by this definition. Changing the field on a $\mu$-null set does not change any induced operator $\pi (g)$.

The measurable field is required to be weakly measurable in each fixed group coordinate; no joint measurability in $(x,g)$ is asserted. The locally compact Hausdorff scope supports the algebraic homomorphism and unitary operators proved above. Strong continuity is a separate assertion, not a consequence claimed here.

Bekka and de la Harpe state the representation construction for second-countable locally compact groups and refer to a separate strong-continuity result. The proof above uses only their fixed-coordinate operator-field construction and the local decomposable-operator theorem, so its algebraic conclusion holds on the stated locally compact Hausdorff scope.