We describe a scheme for constructing quantum mechanics in which the Hilbert space and linear operators are not primary elements of the theory. Some variant of the algebraic approach is instead considered. The elements of a noncommutative algebra (observables) and functionals in this algebra (the elementary states associated with the results of a single measurement) serve as the primary components of the theory. Such a scheme allows us to use, on the one hand, the formalism of the classical (Kolmogorovian) theory of probability, and on the other hand, to reproduce the mathematical formalism of standard quantum mechanics and to specify borders of its applicability. A brief review of necessary data from the theory of algebras and probability theory is given. The manner is described in which the considered mathematical scheme agrees with the theory of quantum measurements and allows one to avoid quantum paradoxes.

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