Axioms required to get Wold-Type decomposition

TL;DR

This paper revisits canonical decompositions of certain operator classes, extending proofs from von Neumann algebras to Baer *-rings using an algebraic approach that avoids topological considerations.

Contribution

It provides an algebraic framework for Wold-type decompositions in Baer *-rings, broadening the scope beyond von Neumann algebra proofs.

Findings

Decompositions valid in Baer *-rings

Algebraic approach avoids topological complexities

Extension of von Neumann algebra results

Abstract

Three canonical decompositions concerning commuting pair of isometries, power partial isometries, and contractions are reassessed. They have already been proved in von Neumann algebras. In the corresponding proofs, both norm and weak operator topologies are heavily involved. Ignoring topological structures, we give an algebraic approach to obtain them in the larger category of Baer $*$-rings.