A new duality via the Haagerup tensor product

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 $C^{\ast }$-modules, it also captures quantum group duality in a fundamental way. We compute the so-called Haagerup dual for various operator algebras arising from ${\mathrm{\ell }}^p$ spaces. In particular, we show that the dual of ${\mathrm{\ell }}^1$ under any operator space structure is $\min {\mathrm{\ell }}^{\mathrm{\infty }}$. 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 $C(\mathbb{G}{)}^h={\mathrm{\ell }}^{\mathrm{\infty }}(\widehat{\mathbb{G}})$, whenever $\widehat{\mathbb{G}}$ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.