Peirce ideals in Jordan algebras

Abstract

In attempting to investigate infinite-dimensional simple Jordan algebras / having rich supplies of idempotents, it would be helpful to know that the Peirce subalgebra Ji(e) relative to an idempotent e in J remains simple. This clearly holds for associative and alternative algebras because any ideal in a Peirce space is the projection of a global ideal. The corresponding result is false for Jordan algebras: there are multiplications of the ambient algebra J which send J to itself (therefore leave invariant the projection of a global ideal), but are not expressible as multiplication by elements of J (therefore need not leave invariant an arbitrary ideal of J). We show that an ideal K x is the projection of a global ideal iff it is invariant under the multiplications Vj 1/2t j 1/2 and UJ 1/2 UJ 1/2 . This yields an explicit expression for the global ideal generated by a Peirce ideal. We then show that if J is a simple Jordan algebra with idempotent, the Peirce subalgebras J and J o inherit simplicity.