Plancherel theorem for unimodular type I groups

Statement

Let $G$ be a second-countable unimodular type I locally compact group with fixed left Haar measure. There is a unique Plancherel measure $\mu_{\mathrm{Pl}}$ on $\widehat G$ such that, for $f\in L^1(G)\cap L^2(G)$ ,

\lVert f\rVert_2^2 =\int_{\widehat G}\lVert\widehat f(\pi)\rVert_{\mathrm{HS}}^2\, d\mu_{\mathrm{Pl}}(\pi).

The nonabelian Fourier transform extends uniquely to a unitary map from $L^2(G)$ onto the direct integral of the Hilbert spaces of Hilbert–Schmidt operators on the representation spaces $\mathcal H_\pi$ . The target inner product is obtained by integrating the Hilbert–Schmidt inner products against $\mu_{\mathrm{Pl}}$ .

Why the hypotheses appear

Second countability supplies the standard measurable framework, and the type I condition makes irreducible direct-integral decomposition essentially unique. Unimodularity removes the Duflo–Moore operators required in the general nonunimodular formula. Thus the theorem is not merely an $L^2$ estimate: its clean operator-valued target depends on all three hypotheses Folland, Theorem 7.50.

Regular-representation disintegration

Under the Plancherel transform, the left regular representation becomes fiberwise left multiplication:

\widehat{\lambda_G(x)f}(\pi)=\pi(x)\widehat f(\pi).

This is the direct-integral decomposition of $\lambda_G$ . Its measure class is supported on the tempered dual, which explains why Plancherel theory sees only irreducibles weakly contained in $\lambda_G$ .

Standard special cases

For an abelian group, the fibers are one-dimensional and the statement is the ordinary LCA Plancherel theorem. For a compact group with Haar probability measure, the integral becomes a sum over the discrete unitary dual with weight $\dim\pi$ , yielding the Peter–Weyl Parseval formula. These examples expose the same theorem with scalar and finite-dimensional operator fibers, respectively.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Theorem 7.50 and the preceding construction of Plancherel measure.
Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: §18.8 on the regular representation, reduced dual, and Plancherel decomposition.