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Compact tripotents in bi-dual JB\(^*\)-triples. (English) Zbl 0853.46070
Summary: The set \({\mathcal U}(C)^\sim\) consisting of the partially ordered set \({\mathcal U}(C)\) of tripotents in a \(\text{JBW}^*\)-triple \(C\) with a greatest element adjoined forms a complete lattice. This paper is mainly concerned with the situation in which \(C\) is the second dual \(A^{**}\) of a complex Banach space \(A\) and, more particularly, when \(A\) is itself a \(\text{JB}^*\)-triple. A subset \({\mathcal U}_c(A)^\sim\) of \({\mathcal U}(A^{**})^\sim\) consisting of the set \({\mathcal U}_c(A)\) of tripotents compact relative to \(A\) (defined in Section 4) with a greatest element adjoined is studied.
It is shown to be an atomic complete lattice with the properties that the infimum of an arbitrary family of elements of \({\mathcal U}_c(A)^\sim\) is the same whether taken in \({\mathcal U}_c(A)^\sim\) or in \({\mathcal U}(A^{**})^\sim\) and that every decreasing net of non-zero elements of \({\mathcal U}_c(A)^\sim\) has a non-zero infimum. The relationship between the complete lattice \({\mathcal U}_c(A)^\sim\) and the complete lattice \({\mathcal U}_c(B)^\sim\), where \(B\) is a Banach space such that \(B^{**}\) is a weak\(^*\)-closed subtriple of \(A^{**}\) is also investigated. When applied to the special case in which \(A\) is a \(C^*\)-algebra the results provide information about the set of compact partial isometries relative to \(A\) and are closely related to those recently obtained by Akemann and Pedersen.
In particular, it is shown that a partial isometry is compact relative to \(A\) if and only if, in their terminology, it belongs locally to \(A\). The main results are applied to this and other examples.

MSC:
46L70 Nonassociative selfadjoint operator algebras
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