Positivstellens\"atze for Algebras of Matrices

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 $\cA$.