Proposition 2

Let A be an involutive Banach algebra, let B be a stellar algebra, and let $\pi$ be a morphism of involutive algebras from A into B. One has $\|\pi (x)\|\leqslant \|x\|$ for all $x\in A$, and in particular $\pi$ is continuous.

For every $x\in A$, one has Sp$'_B(\pi (x))\subset$ Sp$'_A(x)$, therefore $\varrho (\pi (x))\leqslant \varrho (x)\leqslant$ $\|x\|$. Since $\pi (x^*x)\in B_h$, one has $\|\pi (x^*x)\|=\varrho (\pi (x^*x))$ (formula (2)), therefore