A Kronecker Factorization Theorem for Cayley Algebras and the Exceptional Simple Jordan Algebra

Abstract

A well known theorem, due to Wedderburn, states that if $\mathfrak{B}$ is an arbitrary associative algebra with an identity 1 and $\mathfrak{A}$ is a finite dimensional central simple subalgebra containing 1, then $\mathfrak{B}$ is the Kronecker product $\mathfrak{B} = \mathfrak{A} \otimes \mathfrak{U}$ where $\mathfrak{U}$ is the subalgebra of elements which commute with every $a\,\varepsilon \,\mathfrak{A}$. Recently Kaplansky indicated that the same result holds for $\mathfrak{B}$ any alternative algebra and $\mathfrak{A}$ an algebra of Cayley numbers.² It is the purpose of the present note to give a new proof of this result and to show that an analogous result holds for $\mathfrak{B}$ any Jordan algebra and $\mathfrak{A}$ an exceptional simple Jordan algebra of finite dimensions.

The results of this paper were obtained while the author held a Guggenheim Memorial Fellowship.