Involution on a group convolution algebra - Imported Knowlpedia Content

Definition

Let $G$ be a locally compact group, let $\mu$ be a left Haar measure, and let $\Delta$ be its modular function, with $\int_G h(xg)\,d\mu(x)=\Delta(g)^{-1}\int_G h(x)\,d\mu(x)$ . For $f\in L^1(G,\mu)$ , the convolution involution is

f^*(x)=\Delta(x^{-1})\,\overline{f(x^{-1})}

for almost every $x\in G$ . Together with $(f*g)(x)=\int_G f(y)g(y^{-1}x)\,d\mu(y)$ , this operation makes $L^1(G,\mu)$ a Banach $*$ -algebra: it is conjugate-linear, $(f^*)^*=f$ , $(f*g)^*=g^* * f^*$ , and $\lVert f^*\rVert_1=\lVert f\rVert_1$ .

Why the modular factor is present

Group inversion need not preserve a left Haar measure. The factor $\Delta(x^{-1})$ is exactly the Radon–Nikodym correction that compensates for this change of measure. It makes the integrated form of a strongly continuous unitary representation satisfy $\pi(f^*)=\pi(f)^*$ , where $\pi(f)=\int_G f(x)\pi(x)\,d\mu(x)$ . This convention and its compatibility with integrated representations are developed in Folland, §§2.4 and 3.1.

The unimodular case

If $G$ is unimodular, then $\Delta=1$ , so the formula reduces to $f^*(x)=\overline{f(x^{-1})}$ . Dropping the modular factor on a nonunimodular group generally destroys both the $L^1$ -isometry and the adjoint identity for integrated representations.

Conventions and scope

References

Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §§2.4 and 3.1 on convolution, involution, and integrated representations.