Noetherian Jordan Banach algebras are finite-dimensional
An associative or alternative algebra is Noetherian if it satisfies the ascending chain condition on left ideals. \textit{A. M. Sinclair} and \textit{A. W. Tullo} [Math. Ann. 211, 151-153 (1974; Zbl 0275.46037)] showed that a complex Noetherian Banach associative algebra is finite-dimensional. This result was extended by the authors to the alternative case. For a Jordan algebra \(J\), the suitable Noetherian condition is the ascending chain condition on inner ideals. In the paper under review, the authors prove that a complex Noetherian Banach Jordan algebra is finite-dimensional. The proof uses fundamental results on structure theory of Jordan algebras such as nilpotence theorem, prime theorem and Goldie theorem for Jordan algebras, all of them due to E. Zelmanov, and powerful techniques of functional analysis. Reviewer's remark: As shown in a recent paper written by this reviewer jointly with other authors, a real Noetherian Banach Jordan algebra \(J\) is not necessarily finite-dimensional. However, if \(J\) is nondegenerate then it has finite capacity.
scientific article; zbMATH DE number 1270885
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