Generalized limits with additional invariance properties and their applications to noncommutative geometry

Let NN be the set of all natural numbers and l∞=l∞(N)l∞=l∞(N) (resp., L∞=L∞(0,∞)L∞=L∞(0,∞)) be the Banach space of all bounded sequences x=(x1,x2,…)x=(x1,x2,…) (resp., (classes of) Lebesgue measurable bounded functions on (0,∞)(0,∞)) with the uniform norm. We work with subsets of the set of all normalized positive functionals on l∞l∞ and L∞L∞ which are used in noncommutative geometry to define various classes of Dixmier traces (singular positive traces) on the ideal M1,∞M1,∞ of compact operators on an infinite-dimensional Hilbert space HH with logarithmic divergence of the partial sums of their singular values. These classes of traces are of importance in noncommutative geometry and with each such a class (say BB) of traces, a subset M1,∞B⊂M1,∞ on which all traces from BB coincide is linked. The main objective of the present article is to give new characterizations of M1,∞B for commonly occurring classes of Dixmier traces. We also answer a question from Benameur and Fack (2006) [4] concerning conditions on a Dixmier trace under which Lidskii-type formulae from that article hold.