Proposition 20

Let A be a normed involutive algebra and let $j$ be the canonical morphism of A into Stell(A). For every stellar algebra B and every morphism $\varphi$ of involutive algebras from A into B, there exists a unique morphism $\varphi '$ of stellar algebras from Stell(A) into B such that $\varphi =\varphi '\circ j$.

Let us denote by $x\mapsto \|x\|_*$ the norm on Stell(A). Let R be the kernel of $j$. The mapping $x\mapsto \|\varphi (x)\|$ is a stellar seminorm on A. Hence one has $\|\varphi (x)\|\leqslant \|x\|_*$ for all $x\in A$. The morphism $\varphi$ therefore defines, by passing to the quotient, a continuous morphism of $A/R$ into B, which extends by continuity to a morphism $\varphi '$ of Stell(A) into B satisfying $\varphi =\varphi '\circ j$. The uniqueness of $\varphi '$ follows from the fact that the image of $j$ is dense in Stell(A).