Let $A$ be a Banach algebra. Let $L_a,R_a:A\to A$ denote the left/right multiplication operators $$L_ax = ax, \hspace{5mm} R_ax = xa$$ for all $a,x\in A$. Assume that no nonzero $L_a$ and $R_a$ is a compact operator.

Let $\mathcal{M}$ be the set of all bounded linear $T:A\to A^*$ such that $R_a^*T-TL_a$ is compact for every $a\in A$, i.e., the maps $S_a:A\to A^*$ $$S_ax = a(Tx) - T(ax)$$ are compact for each $a\in A$. Clearly the space of compact operators $K(A,A^*)$ is contained in $\mathcal{M}$, and the quotient $\mathcal{M}/{K(A,A^*)}$ is comprised of $A$-module maps ${B(A,A^*)}/{K(A,A^*)}\to {B(A,A^*)}/{K(A,A^*)}$.

Question: I'm looking for conditions on $A$ that are equivalent to $\mathcal{M}=B(A,A^*)$.

Let's also assume that $A$ is unital and reflexive as a Banach space if it simplifies the question. Thanks in advance.

Let $A$ be a Banach algebra. Let $L_a,R_a:A\to A$ denote the left/right multiplication operators $$L_ax = ax, \hspace{5mm} R_ax = xa$$ for all $a,x\in A$. Assume that no nonzero $L_a$ and $R_a$ is a compact operator.

Let $\mathcal{M}$ be the set of all bounded linear $T:A\to A^*$ such that $R_a^*T-TL_a$ is compact for every $a\in A$, i.e., the maps $S_a:A\to A^*$ $$S_ax = a(Tx) - T(ax)$$ are compact for each $a\in A$. Clearly the space of compact operators $K(A,A^*)$ is contained in $\mathcal{M}$.

Question: I'm looking for conditions on $A$ that are equivalent to $\mathcal{M}=B(A,A^*)$.

Let's also assume that $A$ is unital and reflexive as a Banach space if it simplifies the question. Thanks in advance.