COMPACT TRIPOTENTS IN BI-DUAL JB-ASTERISK-TRIPLES

The set U(C)(similar to) consisting of the partially ordered set U(C) of tripotents in a JBW-triple C with a greatest element adjoined form s a complete lattice. This paper is mainly concerned with the situatio n in which C is the second dual A* of a complex Banach space A and, m ore particularly, when A is itself a JB-triple. A subset U-c(A)(simil ar to) of U(A*)(similar to) consisting of the set U-c(A) of tripotent s compact relative to A (defined in Section 4) with a greatest element adjoined is studied. It is shown to be an atomic complete lattice wit h the properties that the infimum of an arbitrary family of elements o f U-c(A)(similar to) is the same whether taken in U-c(A)(similar to) o r in U(A*)(similar to) and that every decreasing net of non-zero elem ents of U-c(A)(similar to) has a non-zero infimum. The relationship be tween the complete lattice U-c(A)(similar to) and the complete lattice U-c(B)(similar to), where B is a Banach space such that B* is a weak -closed subtriple of A** is also investigated. When applied to the sp ecial case in which A is a C-algebra the results provide information about the set of compact partial isometries relative to A and are clos ely related to those recently obtained by Akemann and Pedersen. In par ticular it is shown that a partial isometry is compact relative to A i f and only if, in their terminology, it belongs locally to A. The main results are applied to this and other examples.