On MASAs in $q$-deformed von Neumann algebras

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.