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# Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Published:2017-01-25
Printed: Oct 2017
• Jason Crann,
School of Mathematics and Statistics, Carleton University, Ottawa, ON, Canada K1S 5B6
 Format: LaTeX MathJax PDF

## Abstract

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 $\mathbb{G}$ and 1-injectivity of $L^{\infty}(\widehat{\mathbb{G}})$ as an operator $L^1(\widehat{\mathbb{G}})$-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 $L^1(\widehat{\mathbb{G}})$-module maps on $L^{\infty}(\widehat{\mathbb{G}})$, 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.
 Keywords: locally compact quantum group, amenability, injective module
 MSC Classifications: 22D35 - Duality theorems 46M10 - Projective and injective objects [See also 46A22] 46L89 - Other noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]

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