Semigroup with involution

Abstract: Let S be a semigroup An involutorial anti-isomorphism on S is a map f : S ~ S such that f f ( x ) = x and f ( x y ) = f ( y ) f ( x ) An algebra (S, xy, 2) will be called a semigroup with involution if (S; xy) is a semigroup and 2 is an involutorial antiisomorphism Our aim is to describe the varieties of equationally complete semigroups with involution (For semigroups this was done by Kalicki and Scott [2]) Since semigroups with trivial involution (ie semigroups satisfying 2 = x for every x) are commutative, it follows immediately from the results of [2] that the only equationally complete varieties of semigroups with trivial involution are zero semigroups, semilattices, and Zp (ie varieties of additive abetian groups satisfying the identity px=O, p prime) This last class for p r may also be treated as a class of semigroups with involution 2 = x Obviously this class is also equationally complete There are only two other classes of equationally complete semigroups with involution First is the class of 2-dimensional dice, 3 , introduced and described in [1] ~D is the class of semigroups with involution satisfying identities

Abstract: We study Hermitian kernels invariant under the action of a semigroup with involution. We characterize the Hermitian kernels that realize the given action by bounded operators on a Krein space. Applications include the GNS representation of ∗ -algebras associated to Hermitian functionals and the dilation theory of Hermitian maps on C ∗ -algebras.

Abstract: In a paper from 1976, Berg, Christensen, and Ressel prove that the closure of the cone of sums of squares in the polynomial ring in the topology induced by the -norm is equal to , the cone consisting of all polynomials that are non-negative on the hypercube . The result is deduced as a corollary of a general result, established in the same paper, which is valid for any commutative semigroup. In later work, Berg and Maserick and Berg, Christensen, and Ressel establish an even more general result, for a commutative semigroup with involution, for the closure of the cone of sums of squares of symmetric elements in the weighted -seminorm topology associated with an absolute value. In this paper we give a new proof of these results, which is based on Jacobi’s representation theorem from 2001. At the same time, we use Jacobi’s representation theorem to extend these results from sums of squares to sums of -powers, proving, in particular, that for any integer , the closure of the cone of sums of -powers in the topology induced by the -norm is equal to .