Algebraic representations of von Neumann algebras

TL;DR

This paper introduces an algebraic framework using bilinear Hilbert semispaces to represent von Neumann algebras, linking them with Langlands program concepts and proposing a new classification approach for von Neumann factors.

Contribution

It develops an algebraic extended bilinear Hilbert semispace as a natural representation space for von Neumann algebras, connecting algebraic structures with the Langlands program.

Findings

Constructed towers of von Neumann bisemialgebras algebraically.

Established a correspondence between subbisemimodules and cuspidal representations.

Proposed an algebraic classification method for von Neumann factors.

Abstract

An algebraic extended bilinear Hilbert semispace is proposed as being the natural representation space for the algebras of von Neumann.This bilinear Hilbert semispace has a well defined structure given by the representation space of an algebraic general bilinear semigroup over the product of sets of archimedean completions characterized by increasing degrees.This representation space,decomposing into subbisemimodules according to the pseudounramified or pseudoramified conjugacy classes,is in one-to-one correspondence with the corresponding cuspidal representation according to the Langlands global program.In this context,towers of von Neumann bisemialgebras on the graded bilinear Hilbert semispaces are constructed algebraically which allows to envisage the classification of the factors of von Neumann from an algebraic point of view.