Morita theory for dynamical von Neumann algebras

Abstract

Given a locally compact quantum group $\mathbb{G}$ and two $\mathbb{G}$-$W^{\ast }$-algebras $\alpha :A\curvearrowleft \mathbb{G}$ and $\beta :B\curvearrowleft \mathbb{G}$, we study the notion of equivariant $W^{\ast }$-Morita equivalence $(A,\alpha ){\sim }_{\mathbb{G}}(B,\beta )$, which is an equivariant version of Rieffel's notion of $W^{\ast }$-Morita equivalence. We prove that important dynamical properties of $\mathbb{G}$-$W^{\ast }$-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra $L^{\infty }(\mathbb{K}\backslash \mathbb{G})\subseteq L^{\infty }(\mathbb{G})$ with dual coideal von Neumann algebra $L^{\infty }(\check{\mathbb{K}})\subseteq L^{\infty }(\check{\mathbb{G}})$, we use a natural $\check{\mathbb{G}}$-$W^{\ast }$-Morita equivalence $L^{\infty }(\mathbb{K}\backslash \mathbb{G}){⋊}_{\Delta }\mathbb{G}{\sim }_{\check{\mathbb{G}}}{L}^{\infty }(\check{\mathbb{K}})$ to relate dynamical properties of $L^{\infty }(\mathbb{K}\backslash \mathbb{G})$ with dynamical properties of $L^{\infty }(\check{\mathbb{K}})$. We use this to refine some recent results established by Anderson-Sackaney and Khosravi. This refinement allows us to answer a question of Kalantar, Kasprzak, Skalski and Vergnioux, namely that for $\mathbb{H}$ a closed quantum subgroup of the compact quantum group $\mathbb{G}$, coamenability of $\mathbb{H}\backslash \mathbb{G}$ and relative amenability of ${\ell }^{\infty }(\check{\mathbb{H}})$ in ${\ell }^{\infty }(\check{\mathbb{G}})$ are equivalent. Moreover, if $\mathbb{G}$ is compact, we study the relation between $\mathbb{G}$-$W^{\ast }$-Morita equivalence of $(A,\alpha )$ and $(B,\beta )$ and $\mathbb{G}$-$C^{\ast }$-Morita equivalence of the associated $\mathbb{G}$-$C^{\ast }$-algebras $(\mathcal{R}(A),\alpha )$ and $(\mathcal{R}(B),\beta )$ of regular elements.

24 pages + references. Author accepted version, for publication in International Mathematics Research Notices

References in corpus (1)

• A categorical interpretation of Morita equivalence for dynamical von Neumann algebras