The structure theory of separable complex L*-algebras was given by Schue in [10] and [11]. In [3] Balachandran makes a study of infinite-dimensional complex topologically simple L*-algebras of classical type and poses the question whether these algebras exhaust the class of all infinite-dimensional complex topologically simple L*-algebras. In this paper we give an affirmative answer by determining all the complex topologically simple infinite-dimensional L*-algebras. The case of the real L*-algebras was studied previously in [4], [9] and [12] also under the separability condition. Applying the result of Balachandran, our result yields the structure theory for real L*-algebras. The main tool used here is the ‘approximation’ of the L*-algebra by topologically simple separable L*-algebras via an ultraproduct construction.
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.
This note is strongly inspired by [1] and [2] and it is devoted to the determination of all the composition algebras satisfying some of the Moufang identities.Let F be a field and A 6 ¼ 0 a nonassociative (i.e., not necessarily associative) F-algebra. Assume n : A À! F is a nondegenerate quadratic form. The algebra A is said to be a composition algebra iffor any x; y 2 A: Here the nondegenerate character of n means that if nð ; Þ is the associate bilinear form given by nðx; yÞ ¼ nðx þ yÞ À nðxÞ À nðyÞ; then 0 is the only element x in A such that nðxÞ ¼ 0 ¼ nðx; AÞ: Obviously, if the characteristic of F is different from 2 then nðxÞ ¼ 1 2 nðx; xÞ; and so n is nondegenerate iff 0 is the only element x such that nðx; AÞ ¼ 0: Notice that we do not impose the existence of a unit element in A: When this occurs the well known generalized Hurwitz Theorem asserts that A is isomorphic to 5891
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