On one-sided division infinite-dimensional normed real algebras (Q1802371)

Let \(F\) be a field of characteristic \(\neq 2\) and \(V\), \(W\) vector spaces over \(F\), each with a nondegenerate symmetric bilinear form \(\langle\cdot\mid\cdot\rangle\). The author calls a triple \({\mathcal C}=(V,{}^-,e)\) a Cayley triad iff \({}^-\) is an involutional isometry of \(V\) and \(e\) in \(V\) with \(\overline{e}=e\). Let \({\mathcal L}(W)\) denote the subspace of those \(T\) in \(\text{End}_ F(W)\) with an adjoint \(T^*\) with respect to \(\langle\cdot\mid \cdot\rangle\). A linear map \(S\) from \(V\) into \({\mathcal L}(W)\) with \(S_ x=S(x)\) is said to be a Cayley homomorphism from \({\mathcal C}\) to \(W\) iff \(S_{\overline{x}}\circ S_ x= \langle x\mid x\rangle 1_ W\), \(S_x^*=S_{\overline{x}}\) for any \(x\) in \(V\) and \(S_e=1_W\), where \(1_W\), denotes the identity on \(W\). E.g. \({\mathcal C}\) is a Cayley triad for every composition algebra \(V\) over \(F\) with symmetric bilinear form \(\langle\cdot \mid \cdot\rangle\) and Cayley involution \({}^-\); moreover, \(x\mapsto(y\mapsto yx)\) is a Cayley homomorphism from \({\mathcal C}\) to \(V\) if \(e\) is the unit of \(V\). Using an orthonormal basis and induction the author proves: if \(H\) is a countably infinite dimensional vector space with a positive definite symmetric bilinear form over a Euclidean field, there are a Cayley triad \({\mathcal H}= (H,{}^-,e)\) and a Cayley homomorphism from \({\mathcal H}\) to \(H\). This result yields: If \(H\) is an infinite dimensional separable real Hilbert space there are a Cayley triad \({\mathcal H}=(H,{}^-,e)\) and a Cayley homomorphism from \({\mathcal H}\) to \(H\). As usual let us call an algebra \(V\) absolute valued iff \(V\) has a multiplicative norm and left division algebra iff \(y\mapsto xy\) is bijective for any \(x\) in \(V\setminus\{0\}\). The last result gives: there are absolute valued real left division algebras of infinite dimension in comparison to \textit{F. B. Wright's} conjecture that any absolute valued real division algebra has finite dimension [Proc. Natl. Acad. Sci. USA 39, 330--332 (1953; Zbl 0050.03103)].