Proposition 22

The canonical mapping $j$ of $L^1(G, \nu )$ into Stell(G) is injective and has dense image.

The image of $j$ is dense by definition of the enveloping star algebra of an involutive normed algebra. Since the left regular representation $\boldsymbol{\gamma }$ is faithful, the injectivity of $j$ follows from the equality $\boldsymbol{\gamma }=\boldsymbol{\gamma }'\circ j$.