Some results on the flexible pre-Hilbertian algebras without zero divisors verifying \(||a^2||=||a||^{2}\)
Let \(A\) be a real algebra whose underlying vector space is endowed with a pre-Hilbert norm preserving squares in the sense that \(\| a^2\|=\| a\|^2\) for all \(a\in A\). Then: (1) If \(A\) is flexible without zero divisors and \(\dim A\leq 4\), it is proved that \(A\) is isomorphic to the reals, the complex numbers, or to some mutation \( H^{(\lambda)}\) of the quaternions, or \({\buildrel{\ast}\over{H}}{}^{(\lambda)}\) of the paraquaternions with \(\lambda\neq{1\over 2}\). (2) If \(A\) is flexible without zero divisors containing a central idempotent and satisfying Osborn's properties, then \(A\) is finite-dimensional and isomorphic to the reals, the complex numbers, some mutation \(H^{(\lambda)}\) of the real division quaternions, mutations \({\buildrel\ast\over{ H}}{}^{(\lambda)}\) of the paraquaternions, mutations \(O^{(\lambda)}\) of the real division octonions, mutations \({\buildrel\ast\over{ O}}{}^{(\lambda)}\) of the paraoctonions with \(\lambda\neq{1\over 2}\). In the final section of the work, its author proves that any pre-Hilbert normed algebra \(A\) with unit \(e\) such that \(\| e\|=1\) is flexible and satisfies \(\| a^2\|=\| a\|^2\) for all \(a\in A\). As a corollary, it is proved that such an algebra is Jordan noncommutative. Furthermore, any pre-Hilbert normed algebra without zero divisor and with unit \(e\) satisfying \(\| e\|=1\) and containing a nonzero associative element is isomorphic to the reals, the field of complex numbers or the quaternion division algebra.
scientific article; zbMATH DE number 5298492