Matrix Ordered Operator Algebras

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.

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