Geometry of $C^{\ast }$-algebras, the bidual of their projective tensor product, and completely bounded module maps

Abstract

Let $\mathcal{A}$ be a $C^{\ast }$-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 $C^{\ast }$-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 ${\mathcal{A}}^{\ast }$ has the Schur property, or, again equivalently, the enveloping von Neumann algebra ${\mathcal{A}}^{\ast \ast }$ 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 $C^{\ast }$-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 $C^{\ast }$-algebras $\mathcal{A}$, we determine precisely the centre of the bidual, namely, $Z({V_{\mathcal{A}}}^{\ast \ast })$ is Banach algebra isomorphic to ${\mathcal{A}}^{\ast \ast }{\otimes }_{eh}{\mathcal{A}}^{\ast \ast }$, where ${\otimes }_{eh}$ denotes the extended Haagerup tensor product.

References in corpus (1)

• Proof of the Ghahramani-Lau conjecture