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Abstract

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 $\mathcal R(A)$ 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 $\mathcal R(A)$ of the range tripotents relative to $A$, whilst 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.