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On Jordan ideals and submodules: Algebraic and analytic aspects
Authors:M. Brešar  E. V. Kissin  V. S. Shulman
Affiliation:(1) Department of Mathematics and Computer Sciences, University of Maribor, Maribor, Yugoslavia;(2) Department of Computing, Communication Technology and Mathematics, London Metropolitan University, London, UK;(3) Department of Higher Mathematics, Vologda Technical University, Vologda, Russia
Abstract:
Let
Open image in new window /></a> </div></figure> be an algebra, and let <em class=X be an arbitrary
Open image in new window /></a> </div></figure>-bimodule. A linear space <em class=Y ? X is called a Jordan
Open image in new window /></a> </div></figure>-submodule if <em class=Ay + yA ∈ Y for all A ∈
Open image in new window /></a> </div></figure> and <em class=y ∈ Y. (For X =
Open image in new window /></a> </div></figure>, this coincides with the notion of a Jordan ideal.) We 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. We also consider symmetrically normed Jordan submodules over <em class=C*-algebras. It turns out that there exist C*-algebras in which not all Jordan ideals are ideals.
Keywords:algebra  ideal  bimodule  Jordan ideal   C*-algebra  symmetrically normed ideal
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