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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Local Jordan algebras

Author: Marvin E. Camburn
Journal: Trans. Amer. Math. Soc. 202 (1975), 41-50
MSC: Primary 17A15
MathSciNet review: 0357522
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Abstract: A local Jordan algebra $ \mathfrak{J}$ is a unital quadratic Jordan algebra in which $ \operatorname{Rad} \mathfrak{J}$ is a maximal ideal, $ \mathfrak{J}/\operatorname{Rad} \mathfrak{J}$ satisfies the DCC, and $ { \cap _k}\operatorname{Rad} {\mathfrak{J}^{(k)}} = 0$ where $ {K^{(n + 1)}} = {U_K}(n){K^{(n)}}$. We show that the completion of a local Jordan algebra is also local Jordan, and if $ \mathfrak{J}$ is a complete local Jordan algebra over a field of characteristic not 2, then either (1) $ \mathfrak{J}$ is a complete completely primary Jordan algebra, (2) $ \mathfrak{J} \cong {\mathfrak{J}_1} \oplus {\mathfrak{J}_2} \oplus S$ where each $ {\mathfrak{J}_i}$ is a completely primary local Jordan algebra, or (3) $ \mathfrak{J} \cong \mathfrak{H}({D_n},{J_a})$ where $ (D,j)$ is either a not associative alternative algebra with involution or a complete semilocal associative algebra with involution.

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Keywords: Quadratic Jordan algebra, radical, completion, Jordan matrix algebra
Article copyright: © Copyright 1975 American Mathematical Society