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 VJ1∕2,J1∕2 and UJ1∕2UJ1∕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 J1 and J0 inherit
simplicity.
