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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
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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 locally compact quantum group, amenability, injective module
MSC Classifications: 22D35, 46M10, 46L89 show english descriptions Duality theorems
Projective and injective objects [See also 46A22]
Other ``noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]
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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