Deformation quantization and noncommutative geometry

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By means of a complex symmetric matrix K, we have a linear isomorphism from the Weyl algebra to the space of all polynomial P(C,2n). We introduce a topology into the space P(C,2n) with respect to which the induced product is smooth, and we can take the completion. The star exponential belongs to the complete space. We investigate the two constructions of the star exponential function of the quadratic forms in P(C,2n). One is given by the path integral methods and the other is given by solving an ordinary differential equation. We investiageted the associated bundle structure over the space of symmetric matrices. We also investigate the noncommutative complex projective space given by convergent star products. In future we will study the extended concept of bundle structure which naturally enters through convergent star product algebras.