Vol. 78, No. 2, 1978

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Peirce ideals in Jordan algebras

Kevin Mor McCrimmon

Vol. 78 (1978), No. 2, 397–414

In attempting to investigate infinite-dimensional simple Jordan algebras J having rich supplies of idempotents, it would be helpful to know that the Peirce subalgebra J1(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 J1 to itself (therefore leave invariant the projection of a global ideal), but are not expressible as multiplication by elements of J1 (therefore need not leave invariant an arbitrary ideal of J1). We show that an ideal K1 is the projection of a global ideal iff it is invariant under the multiplications V J12,J12 and UJ12UJ12. 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 J1 and J0 inherit simplicity.

Mathematical Subject Classification 2000
Primary: 17C10
Received: 20 October 1977
Published: 1 October 1978
Kevin Mor McCrimmon