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 socalled 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), whenever G is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.