Separation properties for positive-definite functions on locally compact quantum groups and for associated von Neumann algebras
Jacek Krajczok, Adam Skalski
TL;DR
The paper develops separation properties for quantum positive-definite functions on locally compact quantum groups and transfers these to separation phenomena in the associated von Neumann algebras. By leveraging the Godement mean and generalized Fourier–Stieltjes algebras $B_S(\mathbb{G})$, it shows that the existence of nets of normalised, finitely supported PD functions that remain uniformly away from zero enforces amenability or Haagerup properties, instead of requiring convergence to the identity. For compact quantum groups of Kac type and not coamenable, it establishes a matrix $\varepsilon$-separation property for $L^{\infty}(\mathbb{G})$, and, in the unimodular discrete setting with bounded irreducibles, equates this property with non-injectivity of the von Neumann algebra; the results yield parallel conclusions for classical groups and connect quantum group approximation properties to operator-algebraic rigidity. Collectively, these insights illuminate the tight interplay between separation properties of quantum PD functions, amenability/coamenability, Haagerup properties, and injectivity of von Neumann algebras, with quantum Herz–Schur multipliers playing a central technical role.
Abstract
Using Godement mean on the Fourier-Stieltjes algebra of a locally compact quantum group we obtain strong separation results for quantum positive-definite functions associated to a subclass of representations, strengthening for example the known relationship between amenability of a discrete quantum group and existence of a net of finitely supported quantum positive-definite functions converging pointwise to $I$. We apply these results to show that von Neumann algebras of unimodular discrete quantum groups enjoy a strong form of non-$w^*$-CPAP, which we call the matrix $ε$-separation property.