*-Doubles and embedding of associative algebras in $\mathbf{B}(\mathcal{H})$ Stanislav Popovych 46L0746K5016S1546L0916W10*-algebraHilbert spaceoperator algebra${C^*}$-algebracompletely bounded homomorphismreducing idealembedding We prove that an associative algebra $\mathcal{A}$ is isomorphic to a subalgebra of a $C^{*}$-algebra if and only if its $*$-double $\mathcal{A} * \mathcal{A}^{*}$ is $*$-isomorphic to a $*$-subalgebra of a $C^{*}$-algebra. In particular each operator algebra is shown to be completely boundedly isomorphic to an operator algebra $\mathcal{B}$ with the greatest $C^{*}$-subalgebra consisting of the multiples of the unit and such that each element in $\mathcal{B}$ is determined by its module up to a scalar multiple. We also study the maximal subalgebras of an operator algebra $\mathcal{A}$ which are mapped into $C^{*}$-algebras under completely bounded faithful representations of $\mathcal{A}$. Indiana University Mathematics Journal 2008 text pdf 10.1512/iumj.2008.57.3422 10.1512/iumj.2008.57.3422 en Indiana Univ. Math. J. 57 (2008) 3443 - 3462 state-of-the-art mathematics http://iumj.org/access/