Lemma 16

Let A be an involutive algebra and let $p$ be a seminorm on A. The following conditions are equivalent :

(i) One has $p(xy)\leqslant p(x)p(y),p(x^*) =p(x)$ and $p(x)^2=p(x^*x)$ for all $x, y\in A$;

(ii) The set R of elements $x$ of A such that $p(x) = 0$ is a two-sided self-adjoint ideal of A, and the seminorm on $A/R$ induced by $p$ makes $A/R$ into a normed involutive algebra whose completion is a stellar algebra;

(iii) There exists a stellar algebra B and a morphism $\varphi$ of involutive algebras from A into B such that $p(x) =\|\varphi (x)\|$ for all $x\in A$.

The implications (i)$=\Rightarrow$ (ii)$=\Rightarrow$ (iii)$=\Rightarrow$ (i) are all elementary.

A seminorm satisfying the conditions of lemma 16 will be called a stellar seminorm on the involutive algebra A.

Let A be a normed involutive algebra and let S be the set of stellar seminorms on A. One has $p(x)\leqslant \|x\|$ for all $x\in A$ and all $p\in S$ (I, p. 104, prop. 2). The mapping $x\mapsto \|x\|_*=$ sup$_{p\in S}p(x)$ is a

stellar seminorm on A. It is the greatest stellar seminorm on A.

Let R be the set of $x\in A$ such that $\|x\|_*= 0$. It is a closed two-sided ideal of A. We denote by Stell(A) the completed stellar algebra of $A/R$ for the norm induced by $x\mapsto \|x\|_*$ (lemma 16, (ii)). The canonical mapping of A into Stell(A) is continuous, its image is dense in Stell(A) and its kernel is equal to R.