On Algebraic $\#$-Cones In Topological Tensor-Algebras, I. Basic Properties and Normality
Gerald Hofmann · Publications of the Research Institute for Mathematical Sciences · 1992
The concept of algebraic \sharp -cones (alg- \sharp cones) in topological tensor-algebras E_\otimes[τ] is introduced. It seems to be useful because the well-known cones such as the cone of positivity E_\otimes^+ , the cone of reflection posilivity (Osterwalder–Schrader cone), and some cones of α -positivity in QFT with an indefinite metric are examples of alg- \sharp cones. It is investigated whether or not the known properties of E_\otimes^+ (e.g., E_\otimes^+ is a proper and generating cone not satisfying the decomposition property) apply to alg- \sharp cones. For proving deeper results, the structure of the elements of alg- \sharp 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- \sharp cones is given. Furthermore, the convex hull of finitely many alg- \sharp cones is also considered.