Matrix ordered operator algebras Kate JuschenkoStanislav Popovych 46L0546L0747L5547L0747L30$*$-algebrafaithful representationArchimedean orderoperator system We study the question when for a given $*$-algebra $\mathcal{A}$ a sequence of cones $C_n\subseteq M_n(\mathcal{A})_{sa}$ can be realized as cones of positive operators in a faithful $*$-representation of $\mathcal{A}$ on a Hilbert space. We present a criterion analogous to Effros-Choi abstract characterization of operator systems. A characterization of operator algebras which are completely boundedly isomorphic to $C\sp*$-algebras is also presented. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3608 10.1512/iumj.2009.58.3608 en Indiana Univ. Math. J. 58 (2009) 1203 - 1218 state-of-the-art mathematics http://iumj.org/access/