Proposition 4

Let A be an involutive Banach algebra.

a) For every Hilbert space E, every morphism of involutive algebras $\varphi : A\rightarrow \mathscr{L}(E)$ and every vector $x\in E$, the linear form defined on A by $\lambda (a) =\langle x|\varphi (a)x\rangle$ is a continuous positive linear form;

b) Let $\lambda \in A'_+$. The mapping $f$ of $A\times A$ into $\mathbf{C}$ defined by $f(a, b) =$ $\lambda (a^*b)$ for every $(a, b)\in A\times A$ is a universally positive kernel on A.

Let us prove a). For every $a\in A$, one has

therefore $\lambda$ is a positive linear form on A; it is continuous since $\varphi$ is continuous (Prop. 2 of I, p. 104).

Let us prove b). The function $f$ is continuous. For every $n\in \mathbf{N}$, every family $(a_i)_{0\leqslant i\leqslant n}$ in A and every family $(t_i)_{0\leqslant i\leqslant n}$ of complex numbers, one has

whence the result (Def. 1 of V, p. 433).