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Peer-reviewedCC BY 4.0Axioms

Bijective Maps Preserving a Mixed Triple Product on ∗-Algebras

  • Sanaa Ahmed Bajri1
  • Mohammad Shane Alam2
  1. 1Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
  2. 2Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
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Abstract

Let ℑ and ℜ be unital complex ∗-algebras, and suppose that ℑ contains a nontrivial projection ℘1. Writing ℘2=ℷℑ−℘1, assume the annihilator conditions ϑℑ℘1=0⇒ϑ=0 and ϑℑ℘2=0⇒ϑ=0. If a bijection Φ:ℑ→ℜ preserves the mixed ternary product [ϑ,ς*∘ϱ]* for all ϑ,ς,ϱ∈ℑ, then Φ is additive. Moreover, if Φ(ℷℑ)=ℷℜ, then Φ is a ∗-preserving ring isomorphism. The result yields corresponding statements for prime ∗-algebras, factor von Neumann algebras, and standard operator algebras. We also show that the identity normalization cannot in general be omitted and indicate several directions for extending the argument to broader preserver problems.

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Cite

BibTeXRIS
BibTeX
@article{bajri2026bijective,
  title = {{Bijective Maps Preserving a Mixed Triple Product on ∗-Algebras}},
  author = {Sanaa Ahmed Bajri and Mohammad Shane Alam},
  journal = {Axioms},
  year = {2026},
  month = oct,
  doi = {10.3390/axioms15100756},
  url = {https://www.mdpi.com/2075-1680/15/10/756},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - Bijective Maps Preserving a Mixed Triple Product on ∗-Algebras
AU  - Sanaa Ahmed Bajri
AU  - Mohammad Shane Alam
PY  - 2026
DA  - 2026/10/10
JO  - Axioms
DO  - 10.3390/axioms15100756
UR  - https://www.mdpi.com/2075-1680/15/10/756
AB  - Let ℑ and ℜ be unital complex ∗-algebras, and suppose that ℑ contains a nontrivial projection ℘1. Writing ℘2=ℷℑ−℘1, assume the annihilator conditions ϑℑ℘1=0⇒ϑ=0 and ϑℑ℘2=0⇒ϑ=0. If a bijection Φ:ℑ→ℜ preserves the mixed ternary product [ϑ,ς*∘ϱ]* for all ϑ,ς,ϱ∈ℑ, then Φ is additive. Moreover, if Φ(ℷℑ)=ℷℜ, then Φ is a ∗-preserving ring isomorphism. The result yields corresponding statements for prime ∗-algebras, factor von Neumann algebras, and standard operator algebras. We also show that the identity normalization cannot in general be omitted and indicate several directions for extending the argument to broader preserver problems.
SN  - 2075-1680
ER  -