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The work of Dykema, Figiel, Weiss, and Wodzicki on the structure ofcommutators showed that arithmetic means play an important role in the study ofoperator ideals, and we explored their role in a multipaper project which wesurvey in this article. We start by presenting the notions of arithmetic meanideals and arithmetic mean at infinity ideals. Then we explore theirconnections with commutator spaces, traces, elementary operators, lattice andsublattice structure of ideals, arithmetic mean ideal cancellation propertiesof first and second order, and softness properties - a term that we introducedbut a notion ubiquitous in the literature on operator ideals. Arithmetic meanclosure of ideals leads us to investigate majorization for infinite sequencesand this in turn leads us to an infinite Schur-Horn majorization theorem whichextends theorems by A. Neumann, by Arveson and Kadison, and by Antezana,Massey, Ruiz and Stojanoff. We also list ten open questions that we encounteredin the development of this material.

A survey on the interplay between arithmetic mean ideals, traces, lattices of operator ideals, and an infinite Schur-Horn majorization theorem