Geometry of C⁎-algebras, and the bidual of their projective tensor product

Abstract

Let A be a C * -algebra, and consider the Banach algebra A⊗ γ A, where ⊗ γ denotes the projective Banach space tensor product; if A is commutative, this is the Varopoulos algebra V A . It has been an open problem for more than 35 years to determine precisely when A ⊗ γ A is Arens regular; cf. [26], [40], [41]. Even the situation for commutative A, in particular the case A = ℓ ∞ , 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 A ⊗ γ A is Arens regular if and only if A has the Phillips property; equivalently, A is scattered and has the Dunford-Pettis Property. A further equivalent condition is that A * has the Schur property, or, again equivalently, the enveloping von Neumann algebra A * * is finite atomic, i.e., a direct sum of matrix algebras. Hence, Arens regularity of A ⊗ γ A is encoded in the geometry of the C * -algebra A. In case A is a von Neumann algebra, we conclude that A ⊗ γ A is Arens regular (if and) only if A is finite-dimensional. We also show that this does not generalize to the class of non-selfadjoint dual (even commutative) operator algebras. Specializing to commutative C * -algebras A, we obtain that V A is Arens regular if and only if A is scattered. In fact, we determine precisely the centre of the bidual, namely, Z(V A * * ) is Banach algebra isomorphic to A * * ⊗ eh A * * , where ⊗ eh denotes the extended Haagerup tensor product. We deduce that V A is strongly Arens irregular (if and) only if A is finite-dimensional. Hence, V A is neither Arens regular nor strongly Arens irregular, if and only if A is non-scattered (as mentioned above, this is new even for the case A = ℓ ∞ ).