Complementary inner ideals in JBW⁎JBW⁎-triples

The annihilator L⊥L⊥ of a subspace L   of a JBW⁎JBW⁎-triple A consists of the elements a in A   for which {LaA} is equal to {0}, the kernel Ker(L)Ker(L) of L consists of those elements a in A   for which {LaL} is equal to {0}, and the inner ideal Inid(L)Inid(L) in A associated with L consists of the elements a in A   for which {aLa} is equal to {0} and {LaA} is contained in L. A weak⁎-closed subspace J is said to be an inner ideal in A   if {JAJ} is contained in J, in which caseA=J⊕J1⊕J⊥,A=J⊕J1⊕J⊥, where J1J1 is the intersection of the kernels of J   and J⊥J⊥. The inner ideal Inid(J)Inid(J) in A associated with a weak⁎-closed inner ideal J in A forms a complementary weak⁎-closed inner ideal to J  . It turns out that Inid(J)Inid(J) is compatible with J   and coincides with Inid(J)∩k(J⊥⊥)⊕MJ⊥Inid(J)∩k(J⊥⊥)⊕MJ⊥. In the case where J is a Peirce inner ideal in A  , by completely identifying Inid(J)Inid(J), it is shown that Inid(J)Inid(J) is a Peirce inner ideal in A   and the inner ideal Inid(Inid(J))Inid(Inid(J)) in A   associated with Inid(J)Inid(J) is equal to J.