Exercise 1, § 2

Let $(A, e)$ be a complex unital algebra.

a) For a linear form $\lambda \in A^*$ such that $\lambda (e) = 1$, the following conditions are equivalent:

(i) For every $x\in$ Ker($\lambda$ ), one has $x^2\in$ Ker($\lambda$ );

(ii) For every $x\in A$, one has $\lambda (x^2) =\lambda (x)^2$;

(iii) The kernel of $\lambda$ is a right ideal of A ;

(iv) The linear form $\lambda$ is a unital morphism from A into $\mathbf{C}$.

(Observe that $\lambda (x-\lambda (x)e) = 0$ for every $x\in A$.)

b) Let $\lambda \in A^*$ be a linear form such that $\lambda (1) = 1$ and such that for every $x\in A$, one has $\lambda (x)\in$ Sp$_A(x)$. Let $x\in$ Ker($\lambda$ ) such that $\lambda (x^2)\not= 0$. The spectrum of $x$ is not bounded in $\mathbf{C}$. (For $n\geqslant 2$ integer, consider the polynomial $p\in \mathbf{C}[X]$ such that $p(z) =\lambda ((ze-x)^n)$ for every $z\in \mathbf{C}$; observe that its roots belong to the spectrum of $x$ and minor their euclidean norm as a function of $|\lambda (x^2)|.$)

c) Suppose that A is a Banach algebra. For a linear form $\lambda$ on A to be a unital morphism from A into $\mathbf{C}$, it is necessary and sufficient that $\lambda (e) = 1$ and that $\lambda (x)\in$ Sp$_A(x)$ for every $x\in A$.