Morita Theory for Dynamical von Neumann Algebras

Given a locally compact quantum group G and two G-W*-algebras α : A G and β : B G, we study the notion of equivariant W*-Morita equivalence (A, α )~ G (B, β ), which is an equivariant version of Rieffel’s notion of W*-Morita equivalence. We prove that important dynamical properties of G-W*-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra L^∞ (K G)⊆ L^∞ (G) with dual coideal von Neumann algebra L^∞ (K)⊆ L^∞ (G), we use a natural G-W*-Morita equivalence L^∞ (K G) _Δ G ~ _G L^∞ (K) to relate dynamical properties of L^∞ (K G) with dynamical properties of L^∞ (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 H a closed quantum subgroup of the compact quantum group G, coamenability of H G and relative amenability of ^∞ (H) in ^∞ (G) are equivalent. Moreover, if G is compact, we study the relation between G-W*-Morita equivalence of (A, α ) and (B, β ) and G-C*-Morita equivalence of the associated G-C*-algebras (R(A), α ) and (R(B), β ) of regular elements.