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.