  A scheme for constructing quantum mechanics is given that does not have
Hilbert space and linear operators as its basic elements. Instead, a version of
algebraic approach is considered. Elements of a noncommutative algebra
(observables) and functionals on this algebra (elementary states) associated
with results of single measurements are used as primary components of the
scheme. On the one hand, it is possible to use within the scheme the formalism
of the standard (Kolmogorov) probability theory, and, on the other hand, it is
possible to reproduce the mathematical formalism of standard quantum mechanics,
and to study the limits of its applicability. A short outline is given of the
necessary material from the theory of algebras and probability theory. It is
described how the mathematical scheme of the paper agrees with the theory of
quantum measurements, and avoids quantum paradoxes.
