Summary: 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 M. W. Liebeck and A. Shalev [Ann. Math. (2) 154, No. 2, 383–406 (2001; Zbl 1003.20014)] for non-abelian finite simple groups and analogous results for projective unitary groups of II\(_1\)-factors.