Range tripotents and order in JBW*-triples
Banach Center Publications
In a JBW * -triple, i.e., a symmetric complex Banach space possessing a predual, the set of tripotents is naturally endowed with a partial order relation. This work is mainly concerned with this partial order relation when restricted to the subset R(A) of tripotents in a JBW * -triple B formed by the range tripotents of the elements of a JB * -subtriple A of B. The aim is to present recent developments obtained for the poset R(A) of the range tripotents relative to A, whilst also providing the
... also providing the necessary account of the general theory of the lattice of tripotents. Although the leitmotiv might be described as seeking to find conditions under which the supremum of a subset of range tripotents relative to A is itself a range tripotent relative to A, other properties are also investigated. Amongst these is the relation between range tripotents and partial isometries and support projections in W * -algebras.  c Instytut Matematyczny PAN, 2010 234 L. OLIVEIRA appears in  , although under a different name, and is akin to the concept of range projections in JBW -algebras. The present work is mainly concerned with the subset R(A) of tripotents in a JBW *triple B formed by the range tripotents of the elements of a JB * -subtriple A of B, when endowed with the partial ordering inherited from the set of all tripotents. The aim is to give an overview of recent developments obtained for the poset R(A) whilst also providing the necessary account of the general theory of the lattice of tripotents as to render this work as self-contained as possible. The content of the remaining two sections is as follows. Section 2 mostly includes classical facts on JB * -triples and JBW * -triples appearing in the bibliography, being the notions of order and orthogonality amongst tripotents and properties concerning these notions outlined. Some well-known facts on JB * -algebras and JBW * -algebras are also included here to facilitate future reference. The definition of the range tripotent of an element a in a JBW * -triple B and the definition of range tripotent relative to a JB * -subtriple A of B are made in Section 3. A crucial result appearing in this section is Lemma 3.3, which identifies the range tripotent of a with its range projection in a particular JBW -algebra contained in B. This lemma leads to establishing some circumstances in which the supremum of a subset of range tripotents relative to A is itself a range tripotent relative to A. As a consequence, it is shown that the weak* limits of a particular kind of increasing sequences in the closed unit ball of A are necessarily range tripotents relative to A. It is also investigated in Section 3 how the range tripotents are mapped under isomorphisms and how they relate to the partial isometries and the support projections in W * -algebras. The last results of this section, namely, Theorem 3.6, Corollary 3.7 and Proposition 3.8, are essentially contained in  .