  Starting from an arbitrary endomorphism $\alpha$ of a unital C*-algebra $A$
we construct a bigger C*-algebra $B$ and extend $\alpha$ onto $B$ in such a way
that the extended endomorphism $\alpha$ has a unital kernel and a hereditary
range, i.e. there exists a unique non-degenerate transfer operator for
$(B,\alpha)$, called the complete transfer operator. The pair $(B,\alpha)$ is
universal with respect to a suitable notion of a covariant representation and
depends on a choice of an ideal in $A$. The construction enables a natural
definition of the crossed product for arbitrary $\alpha$.
