Exercise 23, § 6

Let A be a unital star algebra. We denote by $A'_+$ the vector space of linear forms $f$ on A, not necessarily continuous, such that

("positive linear forms").

a) Let $f\in A'_+$. For every sequence $(x_n)_{n\in\mathbf{N}}$ in $A^{\leqslant 1}_+$, the family $(f(x_n))$ is bounded. (Prove that the series $\sum_if(x_i)y_i$ converges for every positive family $(y_i)$ in $\ell^1(\mathbf{N})$.)

b) The linear mapping $f$ is continuous.

c) A linear form $f$ belongs to $A'_+$ if and only if $f$ is continuous and $\|f\|=f(1)$.

d) Let $x\in A_h$ be a hermitian element. There exists a positive linear form $f\in A'_+$ such that $|f(x)|=\|x\|$. (Use Exercise 7 of I, p. 174).