  We study the question how quickly products of a fixed conjugacy class cover
the entire group in the projective unitary group of the connected component of
the identity of the Calkin algebra, as well as the projective unitary group of
a factor von Neumann algebra of type III. Our result is that the number of
factors that are needed is as small as permitted by the (essential) operator
norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple
groups and analogous results for unitary groups of II_1-factors.
