Definition 3

Let A be an involutive Banach algebra. Let $\lambda \in A'_+$ be a continuous positive linear form on A. A Hilbertian realization of $\lambda$ is by definition a triple $(E, x, \varphi )$ consisting of a Hilbert space E, an element $x$ of E and a morphism of involutive algebras $\varphi$ of A into $\mathscr{L}(E)$, such that $\lambda (b^*a) =\langle \varphi (b)x|\varphi (a)x\rangle$ for every $(a, b)\in A^2$.

If the set of elements $\varphi (a)x$ for $a\in A$ is total in E, one says that $(E, x, \varphi )$ is a cyclic Hilbertian realization of $\lambda$.