Proposition 21

Let A be an involutive Banach algebra and $j$ the canonical morphism of A into Stell(A). The radical of A is contained in the kernel of $j$.

Let $x$ be an element of the radical of A. Then $x^*x$ belongs to the radical of A, and therefore Sp$'_A(x^*x) =\{0\}($I, p. 5, remark 3). Since Sp$'_{Stell(A)}(j(x^*x))\subset$ Sp$'_A(x^*x)$, we therefore have Sp$'_{Stell(A)}(j(x)^*j(x)) =\{0\}$, whence $\|j(x)\|^2=\|j(x)^*j(x)\|=\varrho (j(x)^*j(x)) = 0$ (formula (2) of I, p. 104), and therefore $j(x) = 0$.