Derivations of non-separable C∗-algebras

AbstractThe question of which C∗-algebras have only inner derivations has been considered by a number of authors for 25 years. The separable case is completely solved, so this paper deals only with the non-separable case. In particular, we show that the C∗-tensor product of a von Neumann algebra and an abelian C∗-algebra has only inner derivations. Other special types of C∗-algebras are shown to have only inner derivations as well such as the C∗-tensor product of L(H) (all bounded operators on separable Hilbert space) and any separable C∗-algebra having only inner derivations. Derivations from a smaller C∗-algebra into a larger one are also considered, and this concept is generalized to include derivations between C∗-algebras connected by a ∗-homomorphism. Finally, we consider the general problem of a sequence of linear functionals on a C∗-algebra which converges to zero (in norm) when restricted to any abelian C∗-subalgebra. Does such a sequence converge to zero in norm? The answer is “yes” for normal functionals on L(H), but unknown in general