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⁎-modules, it also captures quantum group duality in a fundamental way. We compute the so-called Haagerup dual for various operator algebras arising from ℓp spaces. In particular, we show that the dual of ℓ1 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∞(Gˆ), whenever Gˆ is weakly amenable. Our techniques comprise a mixture of quantum group theory and the geometry of operator space tensor products.

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