  It is shown that the class of Fredholm operators over an arbitrary unital
$C^{*}$--algebra, which may not admit adjoint ones, can be extended in such a
way that this class of compact operators, used in the definition of the class
of Fredholm operators, contains compact operators both with and without
existence of adjoint ones. The main property of this new class is that a
Fredholm operator which may not admit an adjoint one has a decomposition into a
direct sum of an isomorphism and a finitely generated operator. In the space of
compact operators in the Hilbert space a new IM-topology is defined. In the
case when the $C^{*}$--algebra is a commutative algebra of continuous functions
on a compact space the IM-topology fully describe the set of compact operators
over the $C^{*}$--algebra without assumption of existence bounded adjoint
operators over the algebra. In the revised version of the paper the proof of
the theorem 8 has been added.
