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Let $\mathcal{A}$ be a $C^*$-algebra, and consider the Banach algebra$\mathcal{A} \otimes_\gamma \mathcal{A}$, where $\otimes_\gamma$ denotes theprojective Banach space tensor product; if $\mathcal{A}$ is commutative, thisis the Varopoulos algebra $V_\mathcal{A}$. It has been an open problem for morethan 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 remainedunsolved. We solve this classical question for arbitrary $C^*$-algebras byusing von Neumann algebra and operator space methods, mainly relying onversions of the (commutative and non-commutative) Grothendieck Theorem, and thestructure of completely bounded module maps. Establishing these links allows usto show that $\mathcal{A} \otimes_\gamma \mathcal{A}$ is Arens regular if andonly if $\mathcal{A}$ has the Phillips property; equivalently, $\mathcal{A}$ isscattered and has the Dunford--Pettis Property. A further equivalent conditionis that $\mathcal{A}^*$ has the Schur property, or, again equivalently, theenveloping von Neumann algebra $\mathcal{A}^{**}$ is finite atomic, i.e., adirect 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 extendedHaagerup tensor product.

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