Complementary inner ideals in JBW⁎-triples

AbstractThe annihilator L⊥ of a subspace L of a JBW⁎-triple A consists of the elements a in A for which {LaA} is equal to {0}, the kernel Ker(L) of L consists of those elements a in A for which {LaL} is equal to {0}, and the inner ideal 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⊥, where J1 is the intersection of the kernels of J and J⊥. The inner ideal 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) is compatible with J and coincides with Inid(J)∩k(J⊥⊥)⊕MJ⊥. In the case where J is a Peirce inner ideal in A, by completely identifying Inid(J), it is shown that Inid(J) is a Peirce inner ideal in A and the inner ideal Inid(Inid(J)) in A associated with Inid(J) is equal to J