A new duality via the Haagerup tensor product

Abstract

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 ${\ell }^p$ 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 $C(\mathbb{G}{)}^h={\ell }^{\infty }(\hat{\mathbb{G}})$, whenever $\hat{\mathbb{G}}$ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.

18 pages