2018/09/15 by Matthias Neufang, Neufang, Matthias
Mathematics · #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · pdf · doi:10.48550/arxiv.1809.05772
openalex publication_date 2018/09/15 · openalex created_date 2018/09/27 · openalex updated_date 2026/07/28
Let A be a C^*-algebra, and consider the Banach algebra A ⊗γA, where ⊗γ denotes the projective Banach space tensor product; if A is commutative, this is the Varopoulos algebra VA. It has been an open problem for more than 35 years to determine precisely when A ⊗γA is Arens regular. 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 ⊗γA 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 ⊗γA is encoded in the geometry of the C^*-algebra A. In case A is a von Neumann algebra, we conclude that A ⊗γA is Arens regular (if and) only if A is finite-dimensional. For commutative C^*-algebras A, we determine precisely the centre of the bidual, namely, Z(VA**) is Banach algebra isomorphic to A** ⊗eh A**, where ⊗eh denotes the extended Haagerup tensor product.