Example 4

Let G be a locally compact group. Let $\mathscr{M}^1(G)$ be the Banach algebra of bounded complex measures on G (example 6 of I, p. 19).

The mapping $x\mapsto x^{-1}$ of G onto G transforms every measure $\mu\in$ $\mathscr{M}^1(G)$ into a measure $\check{\mu}\in \mathscr{M}^1(G)$ (INT, VII, p. 12, formula (13)). We denote by $\mu^*$ the complex conjugate measure of $\check{\mu}$. The mapping $\mu\mapsto \check{\mu}$ is an isometric isomorphism of the Banach algebra $\mathscr{M}^1(G)$ onto the Banach algebra $\mathscr{M}^1(G^{\circ})$ (INT, VIII, §3, n$^o1$, cor. of prop. 7), hence $\mu\mapsto \mu^*$ is an isometric involution of the Banach algebra $\mathscr{M}^1(G)$.

The set A of bounded measures admitting a density with respect to a Haar measure is a closed subalgebra of $\mathscr{M}^1(G)$ stable under the involution (cf. INT, VIII, §4, n$^o5$); it does not depend on the choice of a Haar measure.

Let $\nu$ be a left Haar measure on G and let us denote by Δ the modulus ofvolution$Gf$. We endow$\mapsto f^*=\widetilde{f}L\overset{1}{\cdot}(G\Delta^-, \nu_1$, where) with the product$\widetilde{f}(x) =\frac{(f, g)}{f(x^{-1})}\mapsto$for every$f*^{\nu}g$ and the in-$x\in G$. Then the mapping $f\mapsto f\cdot \nu$ is an isomorphism of the involutive algebra $L^1(G, \nu )$ onto A. This isomorphism is isometric. In particular, $L^1(G, \nu )$ is identified with an involutive subalgebra of $\mathscr{M}^1(G)$.