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 $A\hat{\otimes }B$ (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 $C^{\ast }$-algebras $A$ and $B$, the Arens regularity of Banach algebras $A{\otimes }^{h}B$, $A\ot^γ B$, $A\ot^{s} B$ and $A\hat{\otimes }B$ are equivalent, where ${\otimes }^h$, ${\otimes }^{\gamma }$, $\ot^s$ and $\hat{\otimes }$ are the Haagerup, the Banach space projective tensor norm, the Schur tensor norm and the operator space projective tensor norm, respectively.