Projective modules in the classical and quantum functional analysis

Along with the classical version, there are two “quantized” versions for the theory of operator algebra. In these lectures, the fundamental homological notion of a projective module is described in the framework of these three theories. Our initial definitions of the projectivity do not go far away from their prototypes in abstract algebra, however, the principal results concern essentially functional-analytic objects and, as a rule, have no purely algebraic analogues.