Automatic continuity of characters
= Automatic continuity of characters
For a <character of an algebra> $\varphi$ on a complex unital <Banach algebra>, $\varphi(1)=1$ and $\varphi(a)\in\sigma(a)$, so $|\varphi(a)|\leq\lVert a\rVert$ without any prior continuity assumption. Under the usual normalization $\lVert1\rVert=1$, this gives $\lVert\varphi\rVert=1$; for an unnormalized <algebra norm>, only the upper bound is automatic.