Positivstellens\"atze for Algebras of Matrices

TL;DR

This paper develops new noncommutative Positivstellens"atze for matrix algebras over various algebraic structures, extending existing results and providing examples to illustrate their applicability.

Contribution

It introduces novel types of Positivstellens"atze for matrix algebras and demonstrates their validity based on properties of the underlying algebra.

Findings

New Positivstellens"atze are proposed and proved.

Examples show the new Positivstellens"atze occur in practice.

Results connect Positivstellens"atze for matrix algebras to those for the base algebra.

Abstract

The paper is concerned with various types of noncommutative Positivstellens\"atze for the matrix algebra $M_n(\cA)$, where $\cA$ is an algebra of operators acting on a unitary space, a path algebra, a cyclic algebra or a formally real field. Some new types of Positivstellens\"atze are proposed and proved, it is shown by examples that they occur. There are a number of results stating that a type of Positivstellensatz is valid for $M_n(\cA)$ provided that it holds for $\cA$.