Representable extensions of positive functionals and Hermitian Banach *-algebras (Q2317443)

Let $A$ be a $*$-algebra, let $\mathcal{H}$ be a complex Hilbert space, and let $\mathcal{B(H)}$ be the space of bounded linear operators on $\mathcal{H}$. A positive functional $f$ is said to be representable if there exists a cyclic $*$-representation $\pi: A\mapsto \mathcal{B(H)}$ with cyclic vector $\xi\in\mathcal{H}$ such that, for any $a\in A$, one has $f(a) = (\pi(a)\xi\vert \xi)$. Extending some previous results, the authors prove the following result, called Theorem 2.5:\par Let $A$ be a $*$-algebra and let $B$ be a $*$-subalgebra of $A$. For a representable positive functional $f : B\mapsto\mathbb{C}$ the following assertions are equivalent:\par (1) there exists a representable positive functional on $A$ which extends $f$;\par (2) there exists a $C^*$-seminorm $p$ on $A$ such that $f$ is continuous with respect to $p\vert B$. \par If, in addition, $f$ is pure then the following assertion is also equivalent with (1) and (2):\par (3) there exists a pure positive functional on $A$ which extends $f$.