Traces on Operator Ideals and Arithmetic Means

This article investigates the codimension of commutator spaces [I, B(H)] of operator ideals on a separable Hilbert space, i.e., “How many traces can an ideal support?” We conjecture that the codimension can be only zero, one, or infinity. Using the arithmetic mean (am) operations on ideals introduced in [13] and the analogous arithmetic mean operations at infinity (am-∞) that we develop extensively in this article, the conjecture is proven for all ideals not contained in the largest am-∞ stable ideal and not containing the smallest am-stable ideal, for all soft-edged ideals (i.e., I = se(I) = IK(H)) and all soft-complemented ideals (i.e., I = scI = I/K(H)), which include all classical operator ideals we considered. The conjecture is proven also for ideals whose soft-interior, seI, or soft-cover, scI, are am-∞ stable or am-stable and for other classes of ideals. We show that an ideal of trace class operators supports a unique trace (up to scalar multiples) if and only if it is am-∞ stable. For a principal ideal, am-∞ stability is what we call regularity at infinity of the sequence of s-numbers of the generator. We prove for these sequences analogs of several of the characterizations of usual regularity. In the process we apply trace extension methods to two problems on elementary operators studied by V. Shulman. This article presents and extends several of the results announced in PNAS-US [19].