Abstract
We study the question of how quickly products of a fixed conjugacy class in the projective unitary group of a II1-factor von Neumann algebra cover the entire group. Our result is that the number of factors that are needed is essentially as small as permitted by the 1-norm – in analogy to results of Liebeck and Shalev for non-abelian finite simple groups. As an application of the techniques, we prove that every homomorphism from the projective unitary group of a finite factor to a Polish SIN group is continuous – a result which is even new for PU(n). Moreover, we show that the projective unitary group of a II1-factor carries a unique Polish group topology.
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