Calderón Martín, A. J.; Martín González, C. Absolute valued triple systems. (English) Zbl 1208.17004 Int. Math. J. 5, No. 8, 741-753 (2004). A triple system \(T\) over the real or the complex numbers is absolute valued if \(| \langle x,y,z\rangle| = | x|| y|| z| \) where \(\langle x,y,z\rangle \) is the triple product Wikipedia Wolfram MathWorld in \(T\) and \(| \cdot| \) is a norm on \(T\). Clearly every absolute valued algebra gives rise to an absolute valued triple system. The paper under review describes the absolute valued triple systems of dimensions 1, 2, and 4. It turns out that over the reals there are two nonisomorphic one-dimensional systems, while over the complex numbers there is only one. (And there cannot exist such systems over the complex numbers whose dimension is larger than 1 but finite.) The results obtained in the paper are exhaustive in dimensions 2 and 4 as well. We mention only that if \(\dim T=2\) over the real numbers then \(T\) must be isomorphic to the complex numbers with any one of six norms explicitly given. The situation in dimension 4 is more complicated and the list involves more possibilities. Reviewer: Plamen Koshlukov (Campinas) MSC: 17A40 Ternary compositions Keywords:absolute valued triple systems of dimensions 1, 2, and 4 × Cite Format Result Cite Review PDF