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 ⁎ -modules, it also captures quantum group duality in a fundamental way. We compute the so-called Haagerup dual for various operator algebras arising from ℓ p spaces. In particular, we show that the dual of ℓ 1 under any operator space structure is min ⁡ ℓ ∞ . In the setting of abstract harmonic analysis we generalize a result of Varopolous by showing that C ( G ) is an operator algebra under convolution for any compact Kac algebra G . We then prove that the corresponding Haagerup dual C ( G ) h = ℓ ∞ ( G ˆ ) , whenever G ˆ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.