Non-linear traces on semifinite factors and generalized singular values

We introduce non-linear traces of the Choquet type and Sugeno type on a semifinite factor \(\mathcal {M}\) as a non-commutative analog of the Choquet integral and Sugeno integral for non-additive measures. We need a weighted dimension function \(p \mapsto \alpha (\tau (p))\) for projections \(p \in \mathcal {M}\), 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 \(\alpha \) are concave. For the algebras of compact operators and factors of type \(\textrm{II}\), we completely determine the condition that the associated weighted \(L^p\)-spaces for the non-linear traces become quasi-normed spaces in terms of the weight functions \(\alpha \) for any \(0< p < \infty \). We also show that any non-linear trace of the Sugeno type yields a certain metric on the factor. This is an attempt at non-linear and non-commutative integration theory on semifinite factors.