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

TL;DR

This paper characterizes when the projective tensor product of a $C^*$-algebra with itself is Arens regular, linking it to the algebra's geometric properties and providing a complete solution to a longstanding open problem.

Contribution

It provides a complete characterization of Arens regularity for the projective tensor product of $C^*$-algebras, connecting it to properties like the Phillips property and the structure of the algebra's bidual.

Findings

Arens regularity iff the algebra has the Phillips property

For von Neumann algebras, regularity iff finite-dimensional

Center of the bidual for commutative case identified with extended Haagerup tensor product

Abstract

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.