We present a new and simple approach to the theory of multiple operator in-tegrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M. G. Kr...

This article is devoted to studying individual ergodic theorems for subsequential weighted averages on the noncommutative Lp-spaces associated a semifinite von Neumann algebra M. In particular, we establish convergence of these along sequences with density one and certain types block positive lower density, extend known results uniform in sense Brunel Keane.

We study the Gnedin–Kingman graph, which corresponds to Pieri’s rule for monomial basis {Mλ} in algebra QSym of quasisymmetric functions. The paper contains a detailed announcement results concerning classification indecomposable semifinite harmonic functions on graph. For these functions, we also establish multiplicativity property, is an analog Vershik–Kerov ring theorem.

For M a separable, purely infinite von Neumann algebra with almost periodic weight φ, a decomposition of M as a crossed product of a semifinite von Neumann algebra by a trace–scaling action of a countable abelian group is given. Then Takasaki’s continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki’s notion of induced action, but here one induce...