  We initiate the study of a new notion of duality defined with respect to the
module Haagerup tensor product. This notion not only recovers the standard
operator space dual for Hilbert $C^*$-modules, it also captures quantum group
duality in a fundamental way. We compute the so-called Haagerup dual for
various operator algebras arising from $\ell^p$ spaces. In particular, we show
that the dual of $\ell^1$ under any operator space structure is
$\min\ell^\infty$. In the setting of abstract harmonic analysis we generalize a
result of Varopolous by showing that $C(\mathbb{G})$ is an operator algebra
under convolution for any compact Kac algebra $\mathbb{G}$. We then prove that
the corresponding Haagerup dual
$C(\mathbb{G})^h=\ell^\infty(\widehat{\mathbb{G}})$, whenever
$\widehat{\mathbb{G}}$ is weakly amenable. Our techniques comprise a mixture of
quantum group theory and the geometry of operator space tensor products.
