  Positivity in $\ast$-algebras can be defined either algebraically, by
quadratic modules, or analytically, by $\ast$-representations. By the induction
procedure for $\ast$-representations we can lift the analytical notion of
positivity from a $\ast$-subalgebra to the entire $\ast$-algebra. The aim of
this paper is to define and study the induction procedure for quadratic
modules. The main question is when a given quadratic module on the
$\ast$-algebra is induced from its intersection with the $\ast$-subalgebra.
This question is very hard even for the smallest quadratic module (i.e. the set
of all sums of hermitian squares) and will be answered only in very special
cases.
