States and *-representations. II
The author extends to a more general *-semigroup setting his previous results given ibid. 17, 163-171 (1986; Zbl 0624.46035). A \(C^*\)- seminorm on a *-semigroup G is a non-negative function p satisfying p(gh)\(\leq p(g)p(h)\) and \(p(g^*g)=p(g)^ 2\) for all g,h in G. A state on G is a complex valued function f satisfying \(| \sum_{g}c_ gf(g)|^ 2\leq M_ f\sum_{g,h}c_ g\bar c_ hf(h^*g)\) for all complex finite sequences \((c_ g)\) indexed by elements of G. A state f is said to be representable if there is a Hilbert space H and a *- representation S of G on H with a cyclic vector x such that \(f(g)=(S_ gx,x).\) Theorem 1. A state f on a *-semigroup G is representable if and only if there is a \(C^*\)-seminorm p on G such hat \(f(g^*g)\leq M_ fp(g)^ 2\) for all g in G. An operator valued function f:G\(\to B(H)\) is said to be dilatable if \(f(g)=VS_ gV^*\), where \(V\in B(K,H)\) and \(g\to S_ g\) is a *-representation of G on a Hilbert space K. An operator valued state is a map f:G\(\to B(H)\) with \(\| \sum_{g}f(g)x_ g\|^ 2\leq M_ f\sum_{g,h}(f(h^*g)x_ g,x_ h)\) for all finite sequences \((x_ g)\) of elements of H. Theorem 2. An operator valued state f on a *-semigroup G is dilatable if and only if \((f(g^*)x,x)\leq M_ fp(g)^ 2\| x\|^ 2\) for all x in H and g in G. Further the author deals with introduced \(S^*\)- semigroups and gives a general form of states or operator valued states on commutative \(C^*\)-semigroups.
scientific article; zbMATH DE number 4042584