Amenability and Covariant Injectivity of Locally Compact Quantum Groups II

Crann, Jason

Canadian journal of mathematics, Tome 69 (2017) no. 5, pp. 1064-1086

Résumé

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 ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$ as an operator ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$ -module. In particular, a locally compact group $G$ is amenable if and only if its group von Neumann algebra $\text{VN}\left( G \right)$ is 1-injective as an operator module over the Fourier algebra $A\left( G \right)$ . As an application, we provide a decomposability result for completely bounded ${{L}^{1}}\left( \widehat{\mathbb{G}} \right)$ -module maps on ${{L}^{\infty }}\left( \widehat{\mathbb{G}} \right)$ , 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.

Détail

DOI : 10.4153/CJM-2016-031-5

Mots-clés : 22D35, 46M10, 46L89, locally compact quantum group, amenability, injective module