2024/10/22 by Joeri De Ro, De Ro, Joeri · 1 citation
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Operator Algebras (math.OA) #Quantum Algebra (math.QA) #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2410.17407
openalex publication_date 2024/10/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a locally compact quantum group \mathbbG and two \mathbbG-W^*-algebras α: A\curvearrowleft \mathbbG and β: B\curvearrowleft \mathbbG, we study the notion of equivariant W^*-Morita equivalence (A, α)∼_\mathbbG (B, β), which is an equivariant version of Rieffel's notion of W^*-Morita equivalence. We prove that important dynamical properties of \mathbbG-W^*-algebras, such as (inner) amenability, are preserved under equivariant Morita equivalence. For a coideal von Neumann algebra L^∞(\mathbbK\backslash \mathbbG)⊆ L^∞(\mathbbG) with dual coideal von Neumann algebra L^∞(\check\mathbbK)⊆ L^∞(\check\mathbbG), we use a natural \check\mathbbG-W^*-Morita equivalence L^∞(\mathbbK\backslash \mathbbG)\rtimesΔ\mathbbG ∼_\check\mathbbG L^∞(\check\mathbbK) to relate dynamical properties of L^∞(\mathbbK\backslash \mathbbG) with dynamical properties of L^∞(\check\mathbbK). 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 \mathbbG, coamenability of ℍ\backslash \mathbbG and relative amenability of ℓ^∞(\checkℍ) in ℓ^∞(\check\mathbbG) are equivalent. Moreover, if \mathbbG is compact, we study the relation between \mathbbG-W^*-Morita equivalence of (A, α) and (B, β) and \mathbbG-C^*-Morita equivalence of the associated \mathbbG-C^*-algebras (R(A), α) and (R(B), β) of regular elements.