Example 4

The involutive algebra $\mathscr{M}^1(G)$ of bounded measures on a locally compact group (I, p. 99, example 4), endowed with the usual norm (example 6 of I, p. 19), is an involutive Banach algebra. Let $\nu$ be a left Haar measure on G. The involutive Banach algebra $L^1(G, \nu )$ is identified with a closed subalgebra of $\mathscr{M}^1(G)$.