We introduce and study the noncommutative modular function spaces of measurable operators affiliated with a semifinite von Neumann algebra and show that they are complete with respect to their modular. We prove that these spaces satisfy the uniform Opial condition with respect to ρ̃-a.e.-convergence for both the Luxemburg norm and the Amemiya norm. Moreover, these spaces have the uniform Kadec–K...

We define a new topology, weaker than the gap topology, on the space of selfadjoint operators affiliated to a semifinite von Neumann algebra and define the real-valued spectral flow for a continuous path of selfadjoint Breuer-Fredholm operators in terms of a generalization of the winding number. We compare our definition with Phillips’ analytical definition. Furthermore we prove the homotopy in...

A theory of unbounded measures is constructed based on the quantum logics orthogonal projections. As an analogue ring sets, projector ideal proposed. Finite and maximal regarding ideals are described. Analogues a number classical theorems measure found. wide class projection characterized. sufficient conditions found to extend integral entire algebra. The problem describing σ-finite in semifini...

Quantum Euclidean spaces, as Moyal deformations of are the model examples noncompact noncommutative manifold. In this paper, we study quantum space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example semifinite spectral triple nonflat geometric structure. We develop abstract symbol calculus for pseudo-differential operators noncommuting deriv...