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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.