Direct integral of unitary representations

Let $G$ be a second-countable locally compact group, $(X,\mu)$ a standard $\sigma$ -finite measure space, and $(H_x)_{x\in X}$ a measurable field of Hilbert spaces. Suppose $\pi_x$ is a strongly continuous unitary representation of $G$ on $H_x$ , and the action field $(x,g)\mapsto\pi_x(g)$ is measurable. The direct integral representation

\pi=\int_X^\oplus\pi_x\,d\mu(x)

acts on the direct-integral Hilbert space by $(\pi(g)\xi)_x=\pi_x(g)\xi_x$ almost everywhere. Equalities and changes of fibers are understood modulo $\mu$ -null sets.

Fiberwise integrated form

For $f\in C_c(G)$ , the integrated operator is decomposable:

\pi(f)=\int_X^\oplus\pi_x(f)\,d\mu(x).

Thus operator-algebraic questions about $\pi(f)$ can often be reduced to almost-everywhere questions about the fibers. This relation also transports direct-integral decompositions between group representations and representations of the full group $C^*$ -algebra.

Decomposition and multiplicity

For suitable separable type I groups, a unitary representation decomposes over the unitary dual into irreducible fibers with a measurable multiplicity function. The regular representation then yields the nonabelian Plancherel measure. Outside the type I setting, irreducible decompositions can have severe nonuniqueness, so direct integrals are often taken instead over factor representations.

Conventions and scope

“Continuous direct sum” is suggestive but does not mean that $x\mapsto H_x$ or $x\mapsto\pi_x$ is continuous in an ordinary bundle topology; measurability is the defining regularity. A direct sum is the special case of counting measure. The phrase “disintegration” may refer either to constructing this representation from fibers or to a theorem asserting that a given representation admits such a decomposition.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §7.4 on direct-integral decompositions of representations.
Jacques Dixmier, $C^*$ -Algebras, North-Holland, 1977. Publisher record. Relevant: §§8.5 and 18.7 on measurable fields and disintegration.