On the symmetrized cones in test function algebra
When Borchers und Uhlmann formulated the Wightman theory in terms of functionals on the algebra of test functions the further development of the theory was very much dependent on knowledge of structure of the algebra. Similar situations occur in the Euclidean approach where the Euclidean fields appear with possible formulation in terms of Euclidean functionals on the algebra of test functions. Since the Euclidean fields commute, the elements of the algebra in question can be taken symmetric with respect to permutations of the variables. Some of the assumed properties of the Euclidean functionals, namely, positivity and T- positivity are formulated with the aid of particular subsets of the algebra which we denote by K and C respectively. The paper is devoted to the study of this sets and the sets sK, sC obtained from K and C by symmetrization in the variables. It is shown by direct calculations that K, C, sK, sC and \(sK+sC\) are sharp cones. Also their closures in some direct sum topology \(\overline{sK}\), \(\overline{sC}\) and \(\overline{sK+sC}\) are sharp cones. Some ideas of Lassner and Uhlmann are utilized in the proofs.
functionals on the algebra of test functions
scientific article; zbMATH DE number 3890169
scientific article