Theorem 1

Let A be a Banach algebra and let $\chi : A\rightarrow \mathbf{C}$ be an algebra morphism (cf. I, p. 9). Then $\chi$ is continuous, of norm at most 1. If A is unital and if $\chi$ is a unital morphism, then $\chi$ is of norm 1.

Let us prove that $|\chi (x)|\leqslant \|x\|$ for every $x\in$ A. Replacing A, if necessary, by the Banach algebra generated by $x$, we may suppose that A is commutative; then, $\chi \in \mathsf{X}'(A)$. For every $x\in A$, one has $\chi (x)\in$ Sp$'_A(x)$ (I, p. 9, No.$^o7$), hence $|\chi (x)|\leqslant \varrho (x)\leqslant \|x\|($I, p. 28, cor. 5), whence the first assertion.

If, moreover, A is unital, the equality $\chi (1) = 1$ implies that $\|\chi \|\geqslant 1$, whence the required equality.