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On Jordan ideals and submodules: Algebraic and analytic aspects. (English. Russian original) Zbl 1178.46048
Funct. Anal. Appl. 42, No. 3, 220-223 (2008); translation from Funkts. Anal. Prilozh. 42, No. 3, 71-75 (2008).
In this brief communication, the following result is obtained. Let $$\mathcal A$$ be an algebra and let $$X$$ be an arbitrary $$\mathcal A$$-bimodule. A linear space $$Y \subset X$$ is called a Jordan $$\mathcal A$$-submodule if $$Ay + yA \in Y$$ for all $$A \in \mathcal A$$ and $$y \in Y$$. (For $$X = \mathcal A$$, this coincides with the notion of a Jordan ideal.) The authors study conditions under which Jordan submodules are subbimodules. General criteria are given in the purely algebraic situation as well as for the case of Banach bimodules over Banach algebras. Simultaneously, the authors also consider symmetrically normed Jordan submodules over $$C^*$$-algebras. It turns out that there exist $$C^*$$-algebras in which not all Jordan ideals are ideals.
Reviewer: Wei Feng (Beijing)
##### MSC:
 46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) 17C50 Jordan structures associated with other structures 16W10 Rings with involution; Lie, Jordan and other nonassociative structures 46L70 Nonassociative selfadjoint operator algebras
##### Keywords:
Jordan ideal; $$C^*$$-alga; symmetrically normed ideal
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