  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.
