Geometry of $C^*$-algebras, the bidual of their projective tensor product, and completely bounded module maps

Abstract

Let $\mathcal{A}$ be a $C^*$-algebra, and consider the Banach algebra $\mathcal{A} \otimes_γ\mathcal{A}$, where $\otimes_γ$ denotes the projective Banach space tensor product; if $\mathcal{A}$ is commutative, this is the Varopoulos algebra $V_\mathcal{A}$. It has been an open problem for more than 35 years to determine precisely when $\mathcal{A} \otimes_γ\mathcal{A}$ is Arens regular. Even the situation for commutative $\mathcal{A}$, in particular the case $\mathcal{A} = \ell_\infty$, has remained unsolved. We solve this classical question for arbitrary $C^*$-algebras by using von Neumann algebra and operator space methods, mainly relying on versions of the (commutative and non-commutative) Grothendieck Theorem, and the structure of completely bounded module maps. Establishing these links allows us to show that $\mathcal{A} \otimes_γ\mathcal{A}$ is Arens regular if and only if $\mathcal{A}$ has the Phillips property; equivalently, $\mathcal{A}$ is scattered and has the Dunford--Pettis Property. A further equivalent condition is that $\mathcal{A}^*$ has the Schur property, or, again equivalently, the enveloping von Neumann algebra $\mathcal{A}^{**}$ is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of $\mathcal{A} \otimes_γ\mathcal{A}$ is encoded in the geometry of the $C^*$-algebra $\mathcal{A}$. In case $\mathcal{A}$ is a von Neumann algebra, we conclude that $\mathcal{A} \otimes_γ\mathcal{A}$ is Arens regular (if and) only if $\mathcal{A}$ is finite-dimensional. For commutative $C^*$-algebras $\mathcal{A}$, we determine precisely the centre of the bidual, namely, $Z({V_\mathcal{A}}^{**})$ is Banach algebra isomorphic to $\mathcal{A}^{**} \otimes_{eh} \mathcal{A}^{**}$, where $\otimes_{eh}$ denotes the extended Haagerup tensor product.