Integral Equations and Operator Theory

, Volume 74, Issue 1, pp 123–136 | Cite as

Jordan Modules and Jordan Ideals of Reflexive Algebras

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Abstract

Let \({\mathcal{L}}\) be a completely distributive subspace lattice on a Banach space and alg \({\mathcal{L}}\) the associated reflexive algebra. Suppose that the following
$$\mbox{Condition A:}\dim(F/F\wedge F_-)\ne1\;\; \mbox{for all}\;\;F\in\mathcal{L}$$
holds; note that if \({\mathcal{L}}\) is an atomic Boolean subspace lattice, this condition means that every atom of \({\mathcal{L}}\) has dimension at least two. It is shown that every reflexive Jordan Alg \({\mathcal{L}}\) -module is an associative Alg \({\mathcal{L}}\) -module. We give an example which shows that if the Condition A is removed, then the conclusion is not necessarily true. Moreover, we prove that all reflexive Jordan ideals of Alg \({\mathcal{L}}\) are associative ideals in the case that no the Condition A is assumed. The same conclusions hold for weakly closed Jordan modules and weakly closed Jordan ideals if the rank one subalgebra of Alg \({\mathcal{L}}\) is weakly dense in Alg \({\mathcal{L}}\) .

Mathematics Subject Classification (2010)

47L35 17C65 46H10 

Keywords

Jordan modules Jordan ideals reflexivity complete distributivity reflexive algebras 

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Mathematics NanjingUniversity of Aeronautics and AstronauticsNanjingPeople’s Republic of China
  2. 2.Department of Computer Science and TechnologyNanjing University of Aeronautics and AstronauticsNanjingPeople’s Republic of China

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