  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 $\cA$.
