On certain multiplier projections

Let $\text{MCZK}$, denote the multiplier algebra over $\text{CZK}$, the algebra of continuous functions into the compact operators with spectrum the infinite product of two-spheres. We consider multiplier projections in $\text{MCZK}$ of a certain diagonal form. We show that, while for each multiplier projection $Q$ of the special form, we have that $Q(x)\in \text{BH}\setminus \text{KK}$ for all $x\in \prod_{j=1}^\infty S^2$,theidealgeneratedby$Q$in$\MCZK$ might be proper. We further show that the ideal generated by a multiplier projection of the special form is proper if and only if the projection is stably finite.