Let $A$ be a Banach algebra and $Bil(A)$ denote the space of bounded bilinear forms on $A$. $Bil(A)$ is a Banach $A$-bimodule with the module operations \begin{eqnarray*} \beta a(x,y) &:=& \beta(ax,y) \\ a \beta(x,y) &:=& \beta(x,ya) \end{eqnarray*} for each $\beta\in Bil(A)$ and each $a,x,y\in A$. Further, for each $f\in A^{\ast}$, define $$\beta_f(x,y) := f(xy) \hspace{4mm} \forall x,y\in A.$$ The set $\{\beta_f:f\in A^{\ast}\}$ is a bi-submodule of $Bil(A)$.

Consider the following hypotheses:

$A$ is unital.

$A$ is reflexive as Banach space.

$\beta_f$ is

*not*weakly sequentially continuous (wsc) for all $f\in A^{\ast}\backslash\{0\}$.

$\beta \in Bil(A)$ is wsc if $\beta(x_n,y_n)\to 0$ whenever $(x_n)$ and $(y_n)$ are weakly null sequences.

3'. Same as 3 with $x_n=y_n$, i.e., for each $f\in A^{\ast}\backslash\{0\}$ there exists a weakly null $(x_n)$ such that $\displaystyle \lim_{n\to\infty}\beta_f(x_n,x_n)\neq 0$.

- $\{\beta_f:f\in A^{\ast}\}$ is a direct bimodule summand of $Bil(A)$, i.e., there exists another bi-submodule $K$ of $Bil(A)$ such that $Bil(A) = K\oplus \{\beta_f:f\in A^{\ast}\}$.

**Question 1.** Does there exist an infinite dimensional Banach algebra that satisfies *1, 2, 3 (or 3')*?

**Question 2.** Does there exist an infinite dimensional Banach algebra that satisfies *1, 2, 3 (or 3'), 4*?