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The present talk is devoted to the role of differential operators in Kapranov's model of noncommutative algebraic geometry. We introduce noncommutative complete schemes from universal enveloping algebra $\mathcal{U% }\left( \mathfrak{g}_{q}\right) $ of the free nilpotent Lie algebra $% \mathfrak{g}_{q}\left( \mathbf{x}\right) $ of index $q$ generated by the independent variables $\mathbf{x}=\left( x_{1},\ldots ,x_{n}\right) $. The formal spectrum of its Lie-completion can be treated as the affine space $% \mathbb{A}_{q}^{n}$. What are noncommutative projective spaces of this model within M. Artin and J. J. Zang formalism of noncommutative projective schemes remains unclear being a serious problem to be discussed. As a technical tool we involve the formally-radical holomorphic functions in elements of a nilpotent Lie algebra. For each (Zariski) open subset $% U\subseteq X=\text{Spec}\left( \mathbb{C}\left[ \mathbf{x}\right] \right) $ we obtain the precise descriptions of the algebras $\mathcal{O}_{q}\left( U\right) $ of noncommutative sections on $U$ associated with the affine spaces $\mathbb{A}_{q}^{n}$. The obtained result for $q=\infty $ generalizes Kapranov's formula. Provided machinery from noncommutative analysis allows us to introduce the projective NC-spaces $\mathbb{P}_{q}^{n}$ for admissible $q$. Related projective schemes are described in term of the differential operators over twisted sheaves.