Let A be a Banach algebra, $(x_n)$ and $(y_n)$ be two weakly null sequences in A. In this talk, we are looking for an answer to the problem how the sequences $(x_ny_n)$ behave weakly. When A has a special algebra structure (e.g., commutative C*-algebras) and/or special Banach space properties (e.g.,Dunford-Pettis property), then one can show that $(x_ny_n)$ is also weakly null. However, this is not always the case. Surprisingly, there is no general treatment of this problem in the literature, to the best of our knowledge. We also tie this problem in with an unsolved conjecture: Reflexive amenable Banach algebras are finite dimensional.

Our aim is not only to present this problem to a general audience, but also to look for answers and constructive criticism from the experts in the field.