Perturbations of self-adjoint operators in semifinite von Neumann algebras: Kato–Rosenblum theorem

The Spectral Scale of a Self-Adjoint Operator in a Semifinite von Neumann Algebra

We extend Akemann, Anderson, and Weaver’s Spectral Scale definition to include selfadjoint operators from semifinite von Neumann algebras. New illustrations of spectral scales in both the finite and semifinite von Neumann settings are presented. A counterexample to a conjecture made by Akemann concerning normal operators and the geometry of the their perspective spectral scales in the finite se...

Divisible Operators in Von Neumann Algebras

Relativizing an idea from multiplicity theory, we say that an element x of a von Neumann algebra M is n-divisible if W (x) ∩ M unitally contains a factor of type In. We decide the density of the n-divisible operators, for various n, M, and operator topologies. The most sensitive case is σ-strong density in II1 factors, which is closely related to the McDuff property. We make use of Voiculescu’s...

Integer Operators in Finite Von Neumann Algebras

Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by M. Fekete and G. Szegö, see [Fek23,FS55,Sze24]. More concretely, we use results by R. Rumely, see [Rum99], on equidistribution of alg...

Kochen-Specker theorem for von Neumann algebras

The Kochen-Specker theorem has been discussed intensely ever since its original proof in 1967. It is one of the central no-go theorems of quantum theory, showing the non-existence of a certain kind of hidden states models. In this paper, we first offer a new, non-combinatorial proof for quantum systems with a type In factor as algebra of observables, including I∞. Afterwards, we give a proof of...

Rank one perturbations of self-adjoint operators