### Abstract:

An algebraic extended bilinear Hilbert semispace is proposed as being the natural representation space for the algebras of Von Neumann, Towers of Von Neumann bisemialgebras on the bilinear Hilbert subsemispace, of which structures are these subbisemimodules, are constructed algebraically which allows to envisage the classification of the Factors of Von Neumann from an algebraic point of view. We show newest results utilize new variants of the noncommutative theorem for spaces, to generalize many of the results concerning cuter functions of the simpler antisymmetric algebra, to the noncommutative Hardy spaces.
We show that every finitely generated affiliated operators module splits as the direct sum of torsion and torsion – free part. We show that the torsion theory concides with the theory of bounded and unbounded modules. We give the necessary and sufficient conditions for affiliated operators to be semisimple. We present a fixed point property of a locally compact group acting on amenable Von Neumann algebra and extended to semifinite normal trace.
We study Effros – Maréchal topology on the space of all Von Neumann algebras acting on fixed separable Hilbert spaces. We prove that factors of some types form dense subset of Von Neumann algebras. We present the full group algebra with no nontrivial projection and it has a separating family of finite dimensional representations. We show that any Centre – Linear derivation of the subalgebra in the algebra of all measurable operators affiliated with Von Neumann algebra is spatial and generated by the subalgebra consisting of all operators.
We compute that Brown's spectral distribution measure for non – normal elements in a finite Von Neumann algebra with respect to a fixed normal faithful tracial state has a polar decomposition with a Haar unitary and * – free.
We investigate that the flow of weights is a canonical factor from the category of separable factors to the category of evgodic flows. The non – commutative flow of weights is another canonical functor from the category of separable factors to the category of covariant systems of semi – finite Von Neumann algebras equipped with trace scaling one parameter automorphism groups with conjugation as morphisms. The construction of these two functors are very similar. The flow of weights functor is obtained by looking at all semi – finite normal weights on a factor with the Murray – Von Neumann equivalence relation the non – commutative flow of weights functor is obtained by relating an arbitrary pair of faithful semi – finite normal weights by the Connes Cocycle. We show the extending of extended modular automorphism of a dominant weight to an arbitrary weight. The construction of the functor ties together the theory of space to the structure theory of a factor of type . Hence the non – commutative flow of weights is obtained by the analytic continuation of space to a pare imaginary value of .
We show that the rigidity for inclusions of finite Von Neumann algebras is equivalent to a meaker property in which no continuity constants are equired. We introduce a new topology on the space of selfadjoint operators affiliated to a semi finite Von Neumann algebra. The real – valued spectral flow for a continuous path of selfadjoint operators is considemened in terms of a generalization of the winding number. We derive integral formula for the spectral flow for certain paths of unbunded operators and the homotopy invariance of the real – valued index.