Invariant Inner Ideals in W * ‐algebras

Let H ( B,α ) be the JBW * ‐algebra of elements of a continuous W * ‐algebra B invariant under the * ‐anti‐automorphism α of B of order two. Then the mapping I → I ∩ H(B, α ) is an order isomorphism from the complete lattice of α‐invariant weak * closed inner ideals in B onto the complete lattice of weak * closed inner ideals in H(B, α ), every one of which is of the form eH(B, α ) α( e ) for some unique projection e in B with α‐invariant central support. A corollary of this result completely characterizes the weak * closed inner ideals in any continuous JBW * ‐triple.