Arens regularity of projective tensor products

Abstract

For completely contractive Banach algebras $A$ and $B$ (respectively operator algebras $A$ and $B$), the necessary and sufficient conditions for the operator space projective tensor product $A\widehat{\otimes}B$ (respectively the Haagerup tensor product $A\otimes^{h}B$) to be Arens regular are obtained. Using the non-commutative Grothendieck's inequality, we show that, for $C^*$-algebras $A$ and $B$, the Arens regularity of Banach algebras $A\otimes^{h}B$, $A\ot^{\gamma} B$, $A\ot^{s} B$ and $A\widehat{\otimes}B$ are equivalent, where $\otimes^h$, $\otimes^{\gamma}$, $\ot^s$ and $\widehat{\otimes}$ are the Haagerup, the Banach space projective tensor norm, the Schur tensor norm and the operator space projective tensor norm, respectively.