# 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

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