Actions of locally compact (quantum) groups on ternary rings of operators, their crossed products and generalized Poisson boundaries

TL;DR

This paper explores how locally compact groups and quantum groups act on W*-ternary rings of operators, introducing crossed products and analyzing fixed point spaces for convolution operators in both classical and quantum settings.

Contribution

It generalizes existing results for von Neumann algebraic actions to the setting of W*-ternary rings, using linking von Neumann algebras for proofs.

Findings

Introduction of crossed products for W*-ternary rings

Generalization of fixed point space analysis to quantum groups

Extension of classical convolution operator results to quantum context

Abstract

Actions of locally compact groups and quantum groups on W*-ternary rings of operators are discussed and related crossed products introduced. The results generalise those for von Neumann algebraic actions with proofs based mostly on passing to the linking von Neumann algebra. They are motivated by the study of fixed point spaces for convolution operators generated by contractive, non-necessarily positive measures, both in the classical and in the quantum context.