@article{Ansari2026, 
author = {Abu Zaid Ansari and Mohammad Shane Alam and Nof T. Alharbi and Ishraga A. Mohamed},
title = {A note on nonlinear mixed bi-skew Jordan and bi-skew Lie    n-derivations on    ∗-algebras},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {6},
pages = {16936-16951},
keywords = {mixed bi-skew Jordan product, bi-skew Lie product, nonlinear derivation, ∗-derivation, prime ∗-algebra, von Neumann algebra},
url = {https://www.sciopen.com/article/10.3934/math.2026693},
doi = {10.3934/math.2026693},
abstract = {Let        A   be a unital    ∗-algebra with identity        I  . For        A    ,      B    ∈      A  , define the bi-skew Jordan product         A    ∙      B    =      A              B              ∗        +      B              A              ∗      and the bi-skew Lie product     [      A    ,      B        ]          ⋄        =      A              B              ∗        −      B              A              ∗        .Suppose that a nonlinear mapping    Φ  :      A    →      A   satisfies     Φ        (                  [                              A                1            ∙                        A                2            ∙      ⋯      ∙                        A                          n          −          1                    ,                        A                n                              ]                          ⋄                      )    =      ∑          j      =      1              n            [              A        1    ∙  ⋯  ∙            A              j      −      1        ∙  Φ  (            A        j    )  ∙            A              j      +      1        ∙  ⋯  ∙            A              n      −      1        ,            A        n              ]              ⋄      for all suitable elements              A        1    ,  …  ,            A        n    ∈      A  , where              A        1    ,            A        2    ∈  {      I    ,  i      I    } and              A        j    =      I   for every    j  =  3  ,  4  ,  …  ,  n  −  2. We prove that    Φ is an additive    ∗-derivation on        A  . As applications, several consequences are obtained for prime    ∗-algebras, factor von Neumann algebras, von Neumann algebras without central summands of type              I        1  , and standard operator algebras. Moreover, a conjecture is proposed to motivate further research in this direction. The obtained results extend and generalize a recent result of Abbasi et al. [Non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on    ∗-algebras, Ricerche Mat., 2026.] concerning non-additive mixed bi-skew Jordan and bi-skew Lie triple derivations on    ∗-algebras.}
}