Let $\mathcal{ A}$ be a $C^*$-algebra, and consider the Banach algebra $\mathcal{ A} \otimes_\gamma \mathcal{ A}$, where $\otimes_\gamma$ 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 $\mathcal{ A} \otimes_\gamma \mathcal{ A}$ 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 $\mathcal{ A} \otimes_\gamma \mathcal{ A}$ 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 $\mathcal{ A} \otimes_\gamma \mathcal{ A}$ is encoded in the geometry of the $C^*$-algebra $\mathcal{ A}$. In case $\mathcal{ A}$ is a von Neumann algebra, we conclude that $\mathcal{ A} \otimes_\gamma \mathcal{ A}$ 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.
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