Bounded Normal Generation for Projective Unitary Groups of Certain Infinite Operator Algebras

Abstract

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 II1-factors.

Strong uncountable cofinality for unitary groups of von Neumann algebras

“…If G is a topologically simple group with neutral element 1 G , then the closure of {1 G } is a closed normal subgroup of G, and hence any nontrivial group topology on G must be Hausdorff. The projective unitary group PU(M ) of M is simple by [17] (this also follows from [10, Theorem 1.2]), respectively [11,Corollary 3.2]. Thus any nontrivial group topology T on PU(M ) is Hausdorff.…”

“…Using Theorem 3.1 we conclude that G has 96-strong uncountable cofinality. If G denotes the projective unitary group of a type III factor, then condition (B) in the above proof also follows from bounded normal generation, see [11,Theorem 1.3], and one can generate G(p) in finitely many steps using the conjugacy class of any element g ∈ G(p) \ {1} and of its inverse. The number of conjugacy classes of g that one needs to generate G(p) then depends on the projective distance of g to the identity in the operator norm.…”

Bounded normal generation and invariant automatic continuity

“…In [5] we show that 40/ℓ ess (•) defines a normal generation function -again optimal up to a multiplicative constant.…”

“…(iii) The connected component PU 1 (C) of the identity of the projective unitary group of the Calkin algebra C has property (BNG), see [5]. A normal generation function is given by…”