  Induced representations of $\ast$-algebras by unbounded operators in Hilbert
space are investigated. Conditional expectations of a $\ast$-algebra $\cA$ onto
a unital $\ast$-subalgebra $\cB$ are introduced and used to define inner
products on the corresponding induced modules. The main part of the paper is
concerned with group graded $\ast$-algebras $\cA=\oplus_{g\in G}\cA_g$ for
which the *-subalgebra $\cB:=\cA_e$ is commutative. Then the canonical
projection $p:\cA\to\cB$ is a conditional expectation and there is a partial
action of the group $G$ on the set $\cBp$ of all characters of $\cB$ which are
nonnegative on the cone $\sum\cA^2\cap\cB.$ The complete Mackey theory is
developed for $\ast$-representations of $\cA$ which are induced from characters
of $\cBp.$ Systems of imprimitivity are defined and two versions of the
imprimitivity theorem are proved in this context. A concept is well-behaved
$\ast$-representations of such $\ast$-algebras $\cA$ is introduced and studied.
It is shown that well-behaved representations are direct sums of cyclic
well-behaved representations and that induced representations of well-behaved
representations are again well-behaved. The theory applies to a large variety
of examples. For important examples such as the Weyl algebra, enveloping
algebras of the Lie algebras $su(2),$ $su(1,1)$, and of the Virasoro algebra,
and $\ast$-algebras generated by dynamical systems our theory is carried out in
great detail.
