On Algebraic $\#$-Cones In Topological Tensor-Algebras, I. Basic Properties and Normality

The concept of algebraic #-cones (alg-$ cones) in topological tensor-algebras E®[T] is introduced. It seems to be useful because the well-known cones such as the cone of positivity E®, the cone of reflection posilivity (Osterwalder-Schrader cone), and some cones of apositivity in QFT with an indefinite metric are examples of alg-# cones. It is investigated whether or not the known properties of £® (e.g., E® is a proper and generating cone not satisfying the decomposition property) apply to alg-# cones. For proving deeper results, the structure of the elements of alg-$ cones is analyzed, and certain estimations between the homogeneous components of those elements are proven. Using them, a detailed investigation of the normality of alg-$ cones is given. Furthermore, the convex hull of finitely many alg-$ cones is also considered. § 0. Introduction The motivation of the present investigations comes from axiomatic quantum field theory (QFT). Within the so-called nonlinear program of the algebraic approach to QFT there are considered several cones in tensor-algebras jE®. Such cones are the cone of positivity E®, [5], [30], the cone of reflection positivity (Osterwalder-Schrader cone), [25], [29], and the convex hull of both, [9]. Furthermore, indefinite inner product QFT and gauge field theories in local (renormalizable) gauges demand some positivity conditions that lead us to the investigation of the cone of a-positivity, [3], [16], [17], [18]. As a generalization of all of these cones the concept of algebraic $-cones (alg-$ cones) is introduced (see Examples 2.4). This series of two papers is devoted to a systematic investigation of the Communicated by H. Araki, June 26, 1991. 1991 Mathematics Subject Classifications: 46K05 * Universitat Leipzig, Fachb. Mathematik, Augustusplatz 10, D-0-7010 Leipzig, Germany