Exercise 3, § 3

a) Give an example of an involutive Banach algebra admitting a positive linear form which is not continuous.

b) Give an example of an involutive Banach algebra A, of a continuous positive linear form $\lambda$ on A, and of Hilbertian realizations $(E_1, e_1, \varphi_1)$ and $(E_2, e_2, \varphi_2)$ of $\lambda$ such that $(E_1, e_1, \varphi_1)$ is a cyclic Hilbertian realization and such that the unique linear mapping $u$ of Prop. 6 of V, p. 440 does not satisfy $u(e_1) =e_2$.