ALGEBRAS OF OPERATORS IN BANACH SPACES OVER THE QUATERNION SKEW FIELD AND THE OCTONION ALGEBRA

摘要

The skew field of quaternions H and the algebra of octonions O are algebras over the real field R, but they are not algebras over the complex field C since all embeddings of C into H or into O are not central. Therefore, the investigation of algebras of operators over H or O cannot be reduced to the study of algebras of operators over C. On the other hand, the theory of algebras of operators over H and O developed below has many specific features in comparison with the general theory of algebras of operators over R owing to the graded structures of the algebras H and O. Moreover, the algebra of octonions cannot be realized as a subalgebra of any algebra of matrices over R since the algebra of octonions is not associative. At the same time, the skew product of the algebra of matrices over C, known as a particular case of the skew product of algebras of operators, produces only algebras over C and cannot give the algebra H or O.