Editing Jordan algebra (section)

==Peirce decomposition==

If ''e'' is an idempotent in a Jordan algebra ''A'' (''e''<sup>2</sup>&nbsp;=&nbsp;''e'') and ''R'' is the operation of multiplication by ''e'', then
* ''R''(2''R''&nbsp;&minus;&nbsp;1)(''R''&nbsp;&minus;&nbsp;1)&nbsp;=&nbsp;0
so the only eigenvalues of ''R'' are 0, 1/2, 1. If the Jordan algebra ''A'' is finite-dimensional over a field of characteristic not 2, this implies that it is a direct sum of subspaces ''A''&nbsp;=&nbsp;''A''<sub>0</sub>(''e'')&nbsp;⊕&nbsp;''A''<sub>1/2</sub>(''e'')&nbsp;⊕&nbsp;''A''<sub>1</sub>(''e'') of the three eigenspaces. This decomposition was first considered by {{harvtxt|Jordan|von Neumann|Wigner|1934}} for totally real Jordan algebras. It was later studied in full generality by {{harvtxt|Albert|1947}}  and called the '''[[Peirce decomposition]]''' of ''A'' relative to the idempotent&nbsp;''e''.<ref>{{harvnb|McCrimmon|2004|pp=99 ''et seq'',235 ''et seq''}}</ref>