completely positive linear maps

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As extensions from function theory to matrix theory and operator theory, there have been fruitful results on integrations, probability, tensor products and many aspects of non-commutative spectral analysis. **Based on some previous results of the proposer, this research work will be carried out on a variety of topics on completely positive linear maps, in connection with numerical ranges and normal dilations, as related to convex structure, spectral variations, matrix synthesis, and harmonic analysis, by means of new approaches in operator theory. In particular, an intensive study is underwaybforn the non-commutative aspects of quantum information. There will also be an investigation on the crossed-products of C*-algebras related to the universal C*-algebras of two isometries/unitaries.