Let $A$ be a unital Banach algebra, $I$ and $J$ be arbitrary index sets, and $P$ be a $J \times I$ nonzero matrix over $A$ such that $\parallel P \parallel_\infty= \{\parallel P_{ij} \parallel:i,j \in I\} \leq 1$. Let $LM(A,P)$ be the vector space of all $I \times J$ matrices $\textit{B}$ over $A$ such that $\parallel B \parallel_1=\sum _{i\in I, j\in J} \parallel A_{i,j} \parallel < \infty$. Then it is easy to check that $LM(A,P)$ with the product $X \circ Y=XPY$, $X,Y \in LM(A,P)$ and the $\ell^1$-norm is a Banach algebra that we call the $\ell^1$--Munn algebra. When $I=J$ and $P$ is the identity $J \times J$ matrix over $A$, we denote $LM(A,P)$ by $M_J(A)$. Gronbaek proved that if $cl(A^2)=A$, then $M_n(A) \otimes_{M_n(A)} M_n(A)\simeq M_n(A \otimes_{A} A)$. Is this true where $n$ replaced by $J$? Since $A$ is unital it is clear that $cl(A^2)=A$.