Geometry of \(C^\ast \)-algebras, and the bidual of their projective tensor product (Q2295689)

It is well known that given a Banach algebra it is possible to extend the multiplication to the bidual in two canonical ways, the left and right Arens multiplication. If both coincide on the bidual, then the algebra is called Arens regular. It is also well known that, for example, C\(^*\)-algebras are Arens regular whereas there are even commutative Arens irregular Banach algebras. The paper under review addresses and finally settles the old question under which geometric conditions on two C\(^*\)-algebras \(A\) and \(B\) their projective tensor product \(A\otimes_\gamma B\) is Arens regular. The answer: This is the case if and only if the dual of \(A\) or of \(B\) has the Schur Property (i.e., weakly convergent sequences converge in norm) which is equivalent (amongst others) to the C\(^*\)-algebra having the Dunford Pettis Property and being scattered (i.e., each positive functional is the weak\(^*\) sum of at most countably many pure states). As a corollary, the projective tensor product of two commutative C\(^*\)-algebras is Arens regular if and only if one of it is scattered because commutative C\(^*\)-algebras have the Dunford Pettis Property. Moreover, if the two C\(^*\)-algebras are von Neumann algebras, then their projective tensor product is Arens regular if and only if one of them is finite dimensional (and, in general, this does not hold for arbitrary Banach algebras: \(\ell_p\otimes_\gamma\ell_p\) (\(1\le p<\infty\), pointwise multiplication on \(\ell_p\)) is Arens regular). The author uses the Grothendieck theorem in order to have equivalence of the projective and the Haagerup tensor product \(\otimes_h\). With the latter, the theory of operator spaces comes in, for example, the author uses the fact that \((E\otimes_h F)^{**}=E^{**}\otimes_{\sigma h} F^{**}\) for two operator spaces \(E\), \(F\), where \(\otimes_{\sigma h}\) is the normal Haagerup tensor product on which a canonical multiplication coming from the identification with the algebra of completely bounded operators (via composition of completely bounded operators) is defined if \(E\) and \(F\) are C\(^*\)-algebras.