# On MASAs in $q$-deformed von Neumann algebras

## TL;DR

This paper investigates $q$-deformed maximal abelian subalgebras in von Neumann algebras, establishing their maximality, singularity, and invariants, and analyzing conjugacy properties among different masas.

## Contribution

It introduces and analyzes $q$-deformed analogues of MASAs, proving their maximality, singularity, and conjugacy properties in various $q$-deformed von Neumann algebras.

## Key findings

- Radial subalgebra is maximal abelian, singular, with Pukánszky invariant {∞}
- All non-equal generator masas are mutually non-unitarily conjugate
- $q$-deformed MASAs exhibit specific structural properties similar to classical cases

## Abstract

We study certain $q$-deformed analogues of the maximal abelian subalgebras of the group von Neumann algebras of free groups. The radial subalgebra is defined for Hecke deformed von Neumann algebras of the Coxeter group $(\mathbb{Z}/{2\mathbb{Z}})^{\star k}$ and shown to be a maximal abelian subalgebra which is singular and with Puk\'anszky invariant $\{\infty\}$. Further all non-equal generator masas in the $q$-deformed Gaussian von Neumann algebras are shown to be mutually non-unitarily conjugate.