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.

Compact quantum groups, Duality, Module Haagerup tensor product, Operator spaces
Journal of Mathematical Analysis and Applications
School of Mathematics and Statistics

Alaghmandan, M. (Mahmood), Crann, J, & Neufang, M. (2018). A new duality via the Haagerup tensor product. Journal of Mathematical Analysis and Applications. doi:10.1016/j.jmaa.2018.11.016