# oa.operator algebras – Is the unit ball of \$A odot B\$ strictly dense in \$M(A otimes B)\$?

Let $$A$$ and $$B$$ be $$C^*$$-algebras and let $$A otimes B$$ their minimal tensor product and $$M(A otimes B)$$ the associated multiplier algebra.

On $$M(A otimes B)$$, we consider the strict topology which is the locally convex topology generated by the seminorms
$$M(Aotimes B)ni xmapsto |x c|$$
$$M(A otimes B)ni xmapsto |cx|$$
for all $$c in Aotimes B$$. This topology is weaker than the norm-topology on $$M(A otimes B)$$.

It is easy to prove that the algebraic tensor product $$A odot B$$ is strictly dense in $$M(A otimes B)$$: simply observe that $$A odot B$$ is norm-dense in $$A otimes B$$ and $$A otimes B$$ is strictly dense in $$M(A otimes B)$$.

Question: Is the unit ball of $$A odot B$$ dense in the unit ball of $$M(A otimes B)$$?