  We prove that C*-algebras which, as Banach spaces, are Grothendieck cannot be
decomposed into a tensor product of two infinite-dimensional C*-algebras. By a
result of Pfitzner, this class contains all von Neumann algebras and their
norm-quotients. We thus complement a recent result of Ghasemi who established a
similar conclusion for the class of SAW*-algebras.
