  We study the question when for a given *-algebra $\mathcal{A}$ a sequence of
cones $C_n\in M_n(\mathcal{A})$ can be realized as cones of positive operators
in a faithful *-representation of $\mathcal{A}$ on a Hilbert space. A
characterization of operator algebras which are completely boundedly isomorphic
to $C\sp*$-algebras is presented.
