# On one-sided division infinite-dimensional normed real algebras

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## Abstract

Publicacions Matemátiques, Vol 36 (1992), 485-488. Abstract ON ONE-SIDED DIVISION INFINITE-DIMENSIONAL NORMED REAL ALGEBRAS JOSÉ ANTONIO CUENCA MIRA Dedicated to the memory of Pere Menal In this note we introduce the concept of Cayley homomorphism which is closely related with those of composition algebra and normalized orthogonal multiplication . The key result shows the existente of certain types of Cayley homomorphisms for infinite dimension. As an application we prove the existente of left division infinite-dimensional complete normed real algebras with left unity . Let K be a field of characteristic different from 2, V and W vector K-spaces (not necessarily with finite dimension) everyone endowed with a nondegenerate symmetric bilinear form. These forms will be denoted by ( 1 ) . We shall say that (V, -, e) is a Cayley triad (over V) if - is an involutive isometry of V and e an element in V such that é = e . We denote by G(W) the vector subspace of the linear maps T in EndK(W) which have an adjoint map T* with respect to ( 1 ) . The linear map S V -j G(W) carrying every x to Sx is said to be a Cayley homomorphism (briefly, homomorphism) from the Cayley triad to W if the following conditions are satisfied : 1) Syo5x = (xix) Id for any x in V (where Id denotes the identity operator on W) . 2) Sx = S, 3) S, = Id . It is easy to show that 2) is equivalent to the following condition : 2') For any x, y, z in V we have (Sx(y)1 Sx(z)) _ (x1x) (y1 z) . Linearizing equality 1) we obtain SxoSy + SgoSx = 2(xly) Id 486 J . A . CUENCA MIRA for any x, y in V . On the other hand 1), 2) and 3) yield to (eje) = 1 . Assume that V = W is a composition K-algebra whose symmetric bilinear form is (1) and with Cayley antiautomorphisrn - . If e is the unity of V then the map sending every element x in V to the rigth multiplication operator R., is a Cayley homomorphism from the Cayley triad (V,-, e) to V. Conversely, let V be a finito-dimensional vector space endowed with a non

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