  We study the *-double functor between the categories of associative and
involutive algebras. It is proved that an associative algebra is isomorphic to
a subalgebra of a $C\sp*$-algebra if and only if its *-double is *-isomorphic
to a *-subalgebra of a $C\sp*$-algebra. Some applications in the theory of
operator algebras are presented. In particular each operator algebra is shown
to be completely boundedly isomorphic to an operator algebra $B$ with the
greatest $C\sp*$-subalgebra consisting of the multiples of the unit and such
that each element in $B$ is determined by its module up to a scalar multiple.
We also study the maximal subalgebras of an operator algebra $A$ which are
mapped into $C\sp*$-algebras under completely bounded faithful representations
of $A$.
