On the symmetrized cones in test function algebra
type: Item
instance of: scholarly article
title: On the symmetrized cones in test function algebra
zbMATH Open document ID: 0558.46038
DOI: 10.1016/0034-4877(80)90091-9
published in: Reports on Mathematical Physics
publication date: 1980-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 46F05
zbMATH DE Number: 3890169
zbMATH Keywords: symmetrized cones
zbMATH Keywords: Wightman theory
zbMATH Keywords: functionals on the algebra of test functions
zbMATH Keywords: Euclidean fields
zbMATH Keywords: Euclidean functionals
zbMATH Keywords: T-positivity
zbMATH Keywords: symmetrization
zbMATH Keywords: sharp cones
author: Witold Karwowski
MaRDI profile type: Publication
full work available at URL: 0034-4877(80)90091-9
OpenAlex ID: W2021626116
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description: scientific article; zbMATH DE number 3890169