TITLE: Continuity of the product map
QUESTION [10 upvotes]: Let $A$ be a $C^*$-algebra. 
Is it possible to characterize $A$ for which the product map defined by
$$\sum\limits_{i=1}^n a_i\otimes b_i \mapsto \sum\limits_{i=1}^n a_i b_i$$
is continuous with respect to the minimal/maximal tensor product of $C^*$-algebras?

REPLY [4 votes]: If you want a reference, see:
MR0806641 (86k:46088)
Quigg, John C.(1-AZS)
Approximately periodic functionals on C∗-algebras and von Neumann algebras.
Canad. J. Math. 37 (1985), no. 5, 769–784.
http://cms.math.ca/10.4153/CJM-1985-043-9
Then Theorem 4.6 shows that for a von Neumann algebra $M$, the multiplication is min continuous if and only if $M$ is type I with each component of bounded finite dimension (i.e. product of matrix algebras with an absolute bound on the dimension).
But Caleb's argument seems more general...