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The Facial and Inner Ideal Structure of a Real JBW*–Triple
Author(s) -
Edwards C. Martin,
Rüttimann Gottfried T.
Publication year - 2001
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200102)222:1<159::aid-mana159>3.0.co;2-n
Subject(s) - mathematics , ideal (ethics) , pure mathematics , epistemology , philosophy
Let B be a real JBW*–triple with predual B * and canonical hermitification the JBW*–triple A It is shown that the set ( B ) ∼ consisting of the partially ordered set ( B ) of tripotents in B with a greatest element adjoined forms a sub–complete lattice of the complete lattice ( A ) ∼ of tripotents in A with the same greatest element adjoined. The complete lattice ( B ) ∼ is shown to be order isomorphic to the complete lattice ℱ n ( B *1 of norm–closed faces of the unit ball B *1 in B * and anti–order isomorphic to the complete lattice ℱ w* ( B 1 ) of weak*–closed faces of the unit ball B 1 in B . Consequently, every proper norm–closed face of B *1 is norm–exposed (by a tripotent) and has the property that it is also a norm–closed face of the closed unit ball in the predual of the hermitification of B . Furthermore, every weak*–closed face of B 1 is weak*–semi–exposed, and, if non–empty, of the form u + B 0 ( u ) 1 where u is a tripotent in B and B 0 ( u ) 1 is the closed unit ball in the zero Peirce space B 0 ( u ) corresponding to u . A structural projection on B is a real linear projection R on B such that, for all elements a and b in B , { Ra b Ra } B is equal to R { a Rb a } B . A subspace J of B is said to be an inner ideal if { J B J } B is contained in J and J is said to be complemented if B is the direct sum of J and the subspace Ker( J ) defined to be the set of elements b in B such that, for all elements a in J , { a b a } B is equal to zero. It is shown that every weak*–closed inner ideal in B is complemented or, equivalently, the range of a structural projection. The results are applied to JBW–algebras, real W*–algebras and certain real Cartan factors.