Bounded normal generation for projective unitary groups of certain infinite operator algebras

TL;DR

This paper investigates how rapidly products of a fixed conjugacy class can generate the entire projective unitary group in certain infinite operator algebras, extending finite group results to infinite-dimensional contexts.

Contribution

It establishes bounds on the number of factors needed to cover the group, matching the minimal operator norm constraints, for specific infinite operator algebra groups.

Findings

Products of conjugacy classes cover the entire group quickly

Bounds are tight and match operator norm limitations

Extends finite group generation results to 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 II_1-factors.