# Journal of Operator Theory

Volume 78, Issue 1,  Summer  2017  pp. 71-88.

Completions of quantum group algebras in certain norms and operators which commute with module actions

Authors:  Mehdi Nemati
Author institution: Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

Summary:  Let $L^1_\mathrm{cb} ({\mathbb G})$ (respectively $L^1_\mathrm M({\mathbb G})$) denote the closure of the quantum group algebra $L^1({\mathbb G})$ of a locally compact quantum group ${\mathbb G}$, in the space of completely bounded (respectively bounded) double centralizers of $L^1({\mathbb G})$. In this paper, we study quantum group generalizations of various results from Fourier algebras of locally compact groups. In particular, left invariant means on $L^1_\mathrm{cb}({\mathbb G})^*$ and on $L^1_\mathrm M({\mathbb G})^*$ are defined and studied. We prove that the set of left invariant means on $L^\infty({\mathbb G})$ and on $L^1_\mathrm{cb}({\mathbb G})^*$ ($L^1_\mathrm M({\mathbb G})^*$) have the same cardinality. We also study the left uniformly continuous functionals associated with these algebras. Finally, for a Banach ${\mathcal A}$-bimodule ${\frak X}$ of a Banach algebra ${\mathcal A}$ we establish a contractive and injective representation from the dual of a left introverted subspace of ${\mathcal A}^*$ into $B_{\mathcal A}({\frak X}^*)$. As an application of this result we show that if the induced representation $\Phi: LUC _\mathrm{cb}({\mathbb G})^*\to B_{L^1_\mathrm{cb}({\mathbb G})}(L^\infty({\mathbb G}))$ is surjective, then $L^1_\mathrm{cb}({\mathbb G})$ has a bounded approximate identity. We also obtain a characterization of co-amenable quantum groups in terms of representations of quantum measure algebras $M({\mathbb G})$.

DOI: http://dx.doi.org/10.7900/jot.2016may30.2120
Keywords:  amenability, Arens regularity, co-amenability, double centralizer, locally compact quantum group