Direct integral of unitary representations - Imported Knowlpedia Content

Definition

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 Folland, §7.4.

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 Dixmier, §18.7.

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.