Non-linear traces on matrix algebras, majorization, unitarily invariant norms and 2-positivity (Q2678413)

There exists a growing interest on the study non-linear positive maps. \textit{T. Ando} and \textit{M.-D. Choi} [in: Nonlinear completely positive maps. Proc. Conf. Occas. H. H. Schaefer's Birthday, Tübingen 1985, North-Holland Math. Stud. 122, 3--13 (1986; Zbl 0606.46035)] and \textit{W. Arveson} [in: Nonlinear states on \(C^*\)-algebras. Proc. Summer Conf., Iowa City/Iowa 1985, Contemp. Math. 62, 283--343 (1987; Zbl 0719.46032)] initiated the study non-linear completely positive maps and extend the Stinespring dilation theorem. \textit{F. Hiai} and \textit{Y. Nakamura} [J. Math. Soc. Japan 39, 367--384 (1987; Zbl 0616.46051)] studied a non-linear counterpart of Arveson's Hahn-Banach type extension theorem for completely positive linear maps. \textit{D. Beltiţá} and \textit{K.-H. Neeb} [Integral Equations Oper. Theory 83, No. 4, 517--562 (2015; Zbl 1331.22017)] studied non-linear completely positive maps and dilation theorems for real involutive algebras. \textit{A. Dadkhah} and \textit{M. S. Moslehian} [Linear Multilinear Algebra 68, No. 8, 1501--1517 (2020; Zbl 07250983)] studied some properties of non-linear positive maps like Lieb maps and the multiplicative domain for 3-positive maps, while \textit{A. Dadkhah} et al. [Stud. Math. 263, No. 3, 241--266 (2022; Zbl 1495.46040)] investigate continuity of non-linear positive maps between C\(^*\)-algebras. The authors of the note under review study several classes of general non-linear positive maps between C\(^*\)-algebras in the recent contribution by \textit{M. Nagisa} and \textit{Y. Watatani} [J. Oper. Theory 87, No. 1, 203--228 (2022; Zbl 07606505)]. A typical construction of non-linear positive maps is given in terms of the functional calculus by a continuous positive function.\par In this interesting paper the authors study non-linear traces of Choquet type and Sugeno type on matrix algebras. Motivated by non-additive measure theory, Choquet integrals and Sugeno integrals can be regarded as non-linear integrals in non-additive measure theory. These two integrals enjoy partial additivities, Choquet integrals have comonotonic additivity and Sugeno integrals have comonotonic F-additivity (fuzzy additivity). Moreover, comonotonic additivity, positive homogeneity and monotony (respectively, comonotonic F-additivity, F-homogeneity and monotony) characterize Choquet integrals (respectively, Sugeno integrals). \par One of the main results, obtained in Theorem 2.7, characterizes non-linear traces of Choquet type on matrix algebras. Namely, for each non-linear positive map \(\varphi: (M_n(\mathbb{C}))^+ \to \mathbb{C}^+\) the following conditions are equivalent:\par (1) \(\varphi\) is a non-linear trace \(\varphi =\varphi_{\alpha}\) of Choquet type associated with a monotone increasing function \(\alpha : \{0, 1, 2, \dots, n\}\to [0,\infty)\) with \(\alpha (0) = 0\).\par (2) \(\varphi\) is monotonic increasing additive on the spectrum, unitarily invariant, monotone and positively homogeneous.\par (3) \(\varphi\) is comonotonic additive on the spectrum, unitarily invariant, monotone and positively homogeneous. \par It is shown in Theorem 3.14 that there exists a close relation among non-linear traces of Choquet type, majorization, unitarily invariant norms and 2-positivity. More concretely, let \(\varphi= \varphi_{\alpha}\) be a non-linear trace of Choquet type associated with a monotone increasing function \(\alpha: \{0, 1, 2, \dots, n\} \to [0,\infty)\) with \(\alpha (0) = 0\) and \(\alpha(1) > 0\). Put \(c_i := \alpha(i) - \alpha (i-1)\) for \(i = 1, 2, \dots, n\) and recall that \(\varphi_{\alpha} = \sum_{i=1}^n c_i \lambda_i\). Define \(\Vert \vert a\Vert \vert_{\alpha}:= \varphi_{\alpha}(\vert a\vert)\) for \(a \in M_n(\mathbb{C})\). Then the following conditions are equivalent:\par (1) \(\alpha\) is concave in the sense that \(\frac{\alpha(i+1)+\alpha(i-1)}{2}\leq \alpha(i)\) (\(i=1, 2, \dots,n-1\)) \par (2) \((c_i)_i\) is a decreasing sequence, i.e., \(c_1 \geq c_2 \geq \dots \geq c_n\).\par (3) \(\Vert \vert \cdot \Vert \vert_{\alpha}\) is a unitarily invariant norm on \(M_n(\mathbb{C})\). \par (4) \(\Vert \vert \cdot\Vert \vert_{\alpha}\) is 2-positive. \par The authors also characterize the non-linear traces of Sugeno type in Theorem 4.4. The ideas are very natural, interesting and motivating.