Let $A$ be a UHF-algebra of type $n^{\infty}$ and denote its unique and faithful trace by $\tau$. Let $L^2(A)$ be the Hilbert space of the GNS-representation associated to $\tau$. We have two commuting representations $L \colon A \to B(L^2(A))$ and $R \colon A^{\rm op} \to B(L^2(A))$ and by the universal property of the maximal tensor product and the nuclearity of $A$, we obtain a $*$-homomorphism $A \otimes A^{\rm op} \to B(L^2(A))$. I think the image of this is weakly dense, i.e. the von Neumann algebra completion of $A \otimes A^{\rm op}$ in this representation is type I and agrees with $B(L^2(A))$. Now consider the $*$-subalgebra$$B = \left\{ \sum_{i} L_{a_i}R_{b_i} \ |\ \sum_{i} a_i b_i = 0\right\} $$spanned by those operators corresponding to elements of $A \otimes A^{\rm op}$ that lie in the kernel of the multiplication map. Let $M$ be the weak closure of $B$ in $B(L^2(A))$.
What "is" this algebra $M$, more precisely: What is the type of $M$? Is it a factor?
