{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:45:36Z","timestamp":1760060736338,"version":"build-2065373602"},"reference-count":17,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University, Deanship of Scientific Research, College of Science Research Center"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Let E and F be two factor von Neumann algebras such that E contains a nontrivial symmetric idempotent element e and an identity element I, with dim(E)\u22652. In this article, we consider a bijective map \u03d1 between E and F satisfying \u03d1(\u03bd1\u2605\u03bd2\u2605\u03bd3\u2605\u22ef\u2605\u03bdn)=\u03d1(\u03bd1)\u2605\u03d1(\u03bd2)\u2605\u03d1(\u03bd3)\u2605\u22ef\u2605\u03d1(\u03bdn) for all \u03bdi\u2208E(i=1,2,\u2026,n), where \u03bdi\u2605\u03bdj=\u03bdi\u2217\u03bdj+\u03bdj\u2217\u03bdi is the bi-skew Jordan product of \u03bdi, \u03bdj for any 1\u2264i,j\u2264n, and n\u22652 is a fixed positive integer. We prove that \u03d1 or \u2212\u03d1 is a conjugate linear \u2217-isomorphism or a linear \u2217-isomorphism. Moreover, for n=2 and n=3, similar results were obtained by Li and Zhang. In this work, we characterize nonlinear bijective maps preserving the n-product for any n\u22652. Thus, our result is more general than both of these earlier results.<\/jats:p>","DOI":"10.3390\/sym17101596","type":"journal-article","created":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T07:46:49Z","timestamp":1758786409000},"page":"1596","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Note on Nonlinear Mappings Preserving the Bi-Skew Jordan-Type Product on Factor von Neumann Algebras"],"prefix":"10.3390","volume":"17","author":[{"given":"Majed","family":"Almubark","sequence":"first","affiliation":[{"name":"Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Asma","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tooba","family":"Naz","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junaid","family":"Nisar","sequence":"additional","affiliation":[{"name":"Department of Applied Sciences, Symbiosis Institute of Technology, Symbiosis International (Deemed) University, Lavale, Pune 412115, India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1090\/S0002-9939-1969-0240129-7","article-title":"When are multiplicative mappings additive","volume":"21","author":"Martindale","year":"1969","journal-title":"Proc. Am. Math. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Ali, A., Naz, T., and Tasleem, M. (Asian-Eur. J. Math., 2025). Nonlinear maps preserving mixed bi-skew Jordan N-product on factor von Neumann algebras, Asian-Eur. J. Math., to appear.","DOI":"10.1142\/S1793557125500585"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.jmaa.2013.07.019","article-title":"Nonlinear maps preserving Jordan \u2217-products","volume":"409","author":"Dai","year":"2014","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2700","DOI":"10.1080\/00927872.2011.584927","article-title":"Additivity of Jordan (triple) derivations on rings","volume":"40","author":"Jing","year":"2012","journal-title":"Commun. Algebra"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"2339","DOI":"10.1016\/j.laa.2012.10.015","article-title":"Nonlinear mappings preserving product XY+YX* on factor von Neumann algebras","volume":"438","author":"Li","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s11785-016-0575-y","article-title":"Nonlinear Maps Preserving the Jordan Triple 1-\u2217-Product on von Neumann Algebras","volume":"11","author":"Li","year":"2017","journal-title":"Complex Anal. Oper. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/S0024-3795(02)00367-1","article-title":"Additivity of Jordan maps on standard operator algebras","volume":"357","author":"Lu","year":"2002","journal-title":"Linear Algebra Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"496","DOI":"10.1215\/20088752-3624940","article-title":"Nonlinear maps preserving the Jordan triple \u2217-product on von Neumann algebras","volume":"7","author":"Li","year":"2016","journal-title":"Ann. Funct. Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1007\/s43034-019-00009-0","article-title":"Additivity of maps preserving Jordan triple products on prime C*-algebras","volume":"11","author":"Taghavi","year":"2020","journal-title":"Ann. Funct. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2159","DOI":"10.1080\/03081087.2016.1142497","article-title":"Maps preserving-Lie product on factor von Neumann algebras","volume":"64","author":"Wang","year":"2016","journal-title":"Linear Multilinear Algebra"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1016\/j.indag.2017.10.010","article-title":"Nonlinear maps preserving the Jordan triple \u2217-product between factors","volume":"29","author":"Zhao","year":"2018","journal-title":"Indag. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.jmaa.2011.07.052","article-title":"Maps preserving products XY-YX* on von Neumann algebras","volume":"386","author":"Bai","year":"2012","journal-title":"J. Math. Anal. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"951","DOI":"10.1080\/03081087.2010.495390","article-title":"Maps preserving product X*Y + YX* on factor von Neumann algebras","volume":"59","author":"Liu","year":"2011","journal-title":"Linear Multilinear Algebra"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"729","DOI":"10.1007\/s41980-018-0048-3","article-title":"Nonlinear maps preserving product X*Y+Y*X on von Neumann algebras","volume":"44","author":"Li","year":"2018","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"757","DOI":"10.1007\/s41980-021-00544-4","article-title":"Maps Preserving Product A*B+B*A on C*-Algebras","volume":"48","author":"Taghavi","year":"2021","journal-title":"Bull. Iran. Math. Soc."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"578","DOI":"10.1007\/s10998-022-00492-4","article-title":"Nonlinear maps preserving bi-skew Jordan triple product on factor von Neumann algebras","volume":"86","author":"Zhang","year":"2023","journal-title":"Period. Math. Hung."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1080\/03081087.2010.533271","article-title":"Additive mappings on C\u2217-algebras preserving absolute values","volume":"60","author":"Taghavi","year":"2012","journal-title":"Linear Multilinear Algebra"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/10\/1596\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:49:17Z","timestamp":1760035757000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/10\/1596"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,25]]},"references-count":17,"journal-issue":{"issue":"10","published-online":{"date-parts":[[2025,10]]}},"alternative-id":["sym17101596"],"URL":"https:\/\/doi.org\/10.3390\/sym17101596","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2025,9,25]]}}}