2015/07/12 by Jason Crann, Crann, Jason
Mathematics · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #math.FA #math.OA
paper · pdf · doi:10.48550/arxiv.1507.03296
Version 4: 21 pages. Results on closed quantum subgroups removed, expanded, and compiled into a separate article
openalex publication_date 2015/07/12 · arxiv created 2016/03/14 · arxiv updated 2016/03/16 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
Building on our previous work, we study the non-relative homology of quantum group convolution algebras. Our main result establishes the equivalence of amenability of a locally compact quantum group \mathbbG and 1-injectivity of L∞(\widehat\mathbbG) as an operator L1(\widehat\mathbbG)-module. In particular, a locally compact group G is amenable if and only if its group von Neumann algebra VN(G) is 1-injective as an operator module over the Fourier algebra A(G). As an application, we provide a decomposability result for completely bounded L1(\widehat\mathbbG)-module maps on L∞(\widehat\mathbbG), and give a simplified proof that amenable discrete quantum groups have co-amenable compact duals which avoids the use of modular theory and the Powers--Størmer inequality, suggesting that our homological techniques may yield a new approach to the open problem of duality between amenability and co-amenability.