Non-commutative generalisations of Urysohn's lemma and hereditary inner ideals
Let \(K\) be a compact Hausdorff space and let \(C \subseteq O\) be subspaces of \(K\) such that \(C\) is closed and \(O\) is open. The classical Urysohn lemma ensures the existence of a positive element \(a\) in the commutative \(C^*\)-algebra \(C(K)\) of all complex-valued continuous functions on \(K\) satisfying \(0 \leq a \leq 1\), \(a\chi_C =\chi_C\) and \(a\chi_{K\setminus O}=0\), where \(\chi\) denotes the characteristic function. A multitude of generalizations of Urysohn's lemma to not necessarily commutative \(C^*\)-algebras have been established during the last forty years. In the paper under review, the authors give several generalizations of Urysohn's lemma to \(JB^*\)-triples which provide full answers to two problems posed by the authors in their paper [``Compact tripotents and the Stone-Weierstrass theorem for \(C^*\)-algebras and \(JB^*\)-triples'', J. Oper. Theory 58, No. 1, 157--173 (2007; Zbl 1150.46027)]. A notion of positively open tripotent is introduced in \(JB^*\)-triples and studied in connection with hereditary inner ideals.
scientific article; zbMATH DE number 5724615
scientific article