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 minℓ∞. 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.
|Keywords||Compact quantum groups, Duality, Module Haagerup tensor product, Operator spaces|
|Journal||Journal of Mathematical Analysis and Applications|
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