Noncommutative weak almost periodicity and left uniform continuity over locally compact quantum groups

Abstract

We study weak almost periodicity and left uniform continuity for Hopf-von Neumann algebras and locally compact quantum groups. For a Hopf-von Neumann algebra $(M,Γ)$, we introduce the space ${\rm WAP}_{{\rm Bessel},π_u}(M_*)$ through norm convergence conditions on rows and columns of slices associated with Bessel families in the universal representation. We prove that this space is the largest $C^*$-subalgebra of ${\rm WAP}(M_*)$ when separately continuous lift of the co-multiplication is multiplicative, and that the two spaces coincide when $M$ is commutative. Furthermore, the algebra $\{x\in C_b(\mathbb{G}): Γ(x)\in M(C_0(\mathbb{G})\otimes C_b(\mathbb{G}))\}$ is characterized as the greatest left introverted $C^*$-subalgebra of ${\rm LUC}(L^1(\mathbb{G}))$ satisfying a left Bessel condition on $L^2(\mathbb{G})$. We also show that ${\rm LUC}(L^1(\mathbb{G}))$ is a $C^*$-algebra whenever the center of $C_0(\mathbb{G})$ contains an element on which the bounded counit is non-zero.