  The geometry of spaces with indefinite inner product, known also as Krein
spaces, is a basic tool for developing Operator Theory therein. In the present
paper we establish a link between this geometry and the algebraic theory of
*-semigroups. It goes via the positive definite functions and related to them
reproducing kernel Hilbert spaces. Our concern is in describing properties of
elements of the semigroup which determine shift operators which serve as
Pontryagin fundamental symmetries
