Nonabelian group Fourier transform

Let $G$ be a locally compact group with a left Haar measure, and let $\widehat G$ be its unitary dual. For $f\in L^1(G)$ , the nonabelian group Fourier transform is the operator-valued field determined weakly by

\langle\widehat f(\pi)\xi,\eta\rangle =\int_G f(g)\langle\pi(g)\xi,\eta\rangle\,dg, \qquad \xi,\eta\in\mathcal H_\pi.

Equivalently, one writes $\widehat f(\pi)=\int_Gf(g)\pi(g)\,dg$ for this integrated operator. It satisfies the operator-norm estimate $\lVert\widehat f(\pi)\rVert\leq\lVert f\rVert_1$ . Replacing $\pi$ by a unitarily equivalent representative conjugates $\widehat f(\pi)$ , so the field is intrinsically defined only up to fiberwise unitary equivalence. For type I groups, the fibers can be organized as a measurable operator field.

Convolution and involution

The transform is the integrated form evaluated at every irreducible representation. Consequently,

\widehat{f*h}(\pi)=\widehat f(\pi)\widehat h(\pi).

With the standard group-algebra involution, it also satisfies $\widehat{f^*}(\pi)=\widehat f(\pi)^*$ . These identities replace scalar diagonalization by simultaneous operator-valued representation of the convolution algebra.

Plancherel realization

For a second-countable unimodular type I group, restricting the transform to $L^1(G)\cap L^2(G)$ gives Hilbert–Schmidt operators for almost every $[\pi]$ and extends to an $L^2$ unitary isomorphism with respect to Plancherel measure. The resulting field simultaneously realizes the direct-integral decomposition of the regular representation.

Conventions and the abelian case

Some authors put $\pi(g^{-1})$ in the integral. That convention matches the usual conjugated-character formula directly but reverses the displayed convolution product unless other conventions are changed. When $G$ is abelian, every fiber is one-dimensional, and after reindexing characters the operator-valued transform becomes the Fourier transform on a locally compact abelian group.

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §§4.1 and 7.4 on group Fourier transforms, integrated representations, and Plancherel theory.
Jacques Dixmier, $C^*$ -Algebras, North-Holland Mathematical Library 15, North-Holland, 1977. Publisher record. Relevant: Chapters 13 and 18 on group $C^*$ -algebras and the dual of a locally compact group.