  In this paper we suggest an approach for constructing an L1-type space for a
positive selfadjoint operator affiliated with von Neumann algebra. For such
operator we intro- duce a seminorm, and prove that it is a norm if and only if
the operator is injective. For this norm we construct an L1 -type space as the
complition of the space of hermitian ultraweakly continuous linear functionals
on von Neumann algebra, and represent L1- type space as a space of continuous
linear functionals on the space of special sesquilinear forms. Also, we prove
that L1 -type space is isometrically isomorphic to the predual of von Neumann
algebra in a natural way. We give a small list of alternate definitions of the
seminorm, and a special definition for the case of semifinite von Neumann
algebra, in particular. We study order properties of L1-type space, and
demonstrate the con- nection between semifinite normal weights and positive
elements of this space. At last, we construct a similar L-space for the
positive element of C*-algebra, and study the connection between this L-space
and the L1 -type space in case when this C*-algebra is a von Neumann algebra.
