Exponentially bounded indefinite functions (Q1113051)

Let \(<,>\) be a hermitian sesquilinear form on a *-algebra A satisfying \(<ab,c>=<b,a^*c>\) for all a,b,c in A. Assume that A contains a subspace of finite codimension which is positive: \(<b,b>\geq 0\) for all b in that subspace. The Gelfand-Naǐmark-Segal construction applied to \(<,>\) yields a *-representation of A by densely defined operators in a Pontrjagin space. It is shown that these operators are bounded if \(<,>\) is bounded with respect to some submultiplicative seminorm. The converse holds if A has an identity. If A is commutative and the operators in the GNS representation are bounded, a theorem of Naǐmark implies that A contains a maximal positive subspace which is an ideal. Applications to functions with finitely many negative squares on commutative *-semigroups are given.