A characterisation of von Neumann regular Jordan triple systems

Meyberg, Kurt

doi:10.1090/S0002-9939-1975-0376795-8

A characterisation of von Neumann regular Jordan triple systems
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Proc. Amer. Math. Soc. 49 (1975), 25-27 Request permission

Abstract:

In this note we give a characterisation of the Jacobson radical of a Jordan triple system in terms of principal inner ideals. If $(\mathfrak {A},P)$ is a Jordan triple system and $\operatorname {Rad} \mathfrak {A}$ the Jacobson radical of $\mathfrak {A}$ then $x \in \operatorname {Rad} \mathfrak {A}$ iff $P(x)\mathfrak {A} = P(x + P(x)y)\mathfrak {A}$ for all $y \in \mathfrak {A}$. We use this to give a new characterisation of von Neumann regular Jordan triple systems. In particular, this gives another most elementary and short proof that semi-simple Jordan triple systems with dcc on principal inner ideals are von Neumann regular.

References

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