Non-Linear Traces on Semifinite Factors and Generalized Singular Numbers

We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor M as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need weighted dimension function p ↦ α(τ(p)) for projections p ∈ M, which is an analog of a monotone measure. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both the Choquet type and Sugeno type respectively. Based on the notion of generalized eigenvalues and singular values, we show that non-linear traces of the Choquet type are closely related to the Lorentz function spaces and the Lorentz operator spaces if the weight functions α are concave. For the algebras of compact operators and factors of type II, we completely determine the condition that the associated weighted Lᵖ-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions α for any 0 < p < ∞. We also show that any non-linear trace of the Sugeno type gives a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors.