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• W2891540726 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." @default.
• W2891540726 created "2018-09-27" @default.
• W2891540726 creator A5087063369 @default.
• W2891540726 date "2018-09-15" @default.
• W2891540726 modified "2023-09-27" @default.
• W2891540726 title "Geometry of $C^*$-algebras, the bidual of their projective tensor product, and completely bounded module maps" @default.
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• W2891540726 hasPublicationYear "2018" @default.
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