On range tripotents in JBW*-triples

Abstract

Let B be a JBW * -triple, let A be a JB * -subtriple of B and let R(A) be the set of range tripotents relative to A. It is shown that, under certain conditions, the supremum of a family of range tripotents in R(A) coincides with that in the complete latticeŨ(B) of all tripotents in B. As a consequence, a sufficient condition for a tripotent to be a range tripotent relative to A is obtained. The action of isomorphisms on range tripotents is investigated, and an analysis of the suprema of families of spectral range tripotents leads to a generalization of a result known for open projections in W * -algebras. Mathematics Subject Classification (2000). Primary 46L70; Secondary 17C65.

Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals

“…The hull-kernel structure in JB * -triples has been intensively developed in recent years (see for example [20,14,21,22,9,17,32] and [24]). In this setting, several versions of Urysohn's lemma were established in [14,Theorem 3.3], [ [22].…”

Range tripotents and order in JBW*-triples

“…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 [19].…”

“…The next lemma establishes a relation between range tripotents and range projections. Although the lemma may be found in [19], its proof is included here for the reader's convenience.…”