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

Abstract

The concept of algebraic $♯$-cones (alg-$♯$ cones) in topological tensor-algebras $E_{\otimes }[\tau ]$ 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 $\alpha$-positivity in QFT with an indefinite metric are examples of alg-$♯$ 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-$♯$ 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.