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Projective module description of embedded noncommutative spaces

An algebraic formulation is given for the embedded noncommutative spaces over the Moyal algebra developed in a geometric framework in \cite{CTZZ}. We explicitly construct the projective modules corresponding to the tangent bundles of the embedded noncommutative spaces, and recover from this algebraic formulation the metric, Levi-Civita connection and related curvatures, which were introduced geometrically in \cite{CTZZ}. Transformation rules for connections and curvatures under general coordinate changes are given. A bar involution on the Moyal algebra is discovered, and its consequences on the noncommutative differential geometry are described.