States and\( ^ *\)-representations. I
The author introduces and studies a concept of an \(S^*\)-algebra. It is a complex *-algebra A posessing a \(C^*\)-seminorm \(p_ A\) such that every state f on A is continuous with respect to \(p_ A\) (a state is a positive functional satisfying \(| f(a)|^ 2\leq M_ ff(a^*a)\) for all a in A). Among other results the author shows that A is an \(S^*\)-algebra if and only if \[ \sup \{f(b^*a^*ab):b\in A,\text{ f is a state and }f(b^*b)\leq 1\}<\infty \] for all a in A, and that all \(C^*\)-seminorms on an \(S^*\)-algebra A are majorized by \(p_ A\). There is also given a general form of completely positive maps from an \(S^*\)-algebra into B(H) and it is shown that every Schwartz map on a commutative \(S^*\)-algebra is automatically completely positive.
scientific article; zbMATH DE number 4013312