Von Neumann regularity and quadratic conorms in JB\(^*\)-triples and C\(^*\)-algebras
For an element \(a\) in a \(\text{JB}^*\)-triple \(A\), the quadratic operator \(Q(a)\) is defined, for each element \(b\) in \(A\), by \[ Q(a)(b)= \{a b a\}, \] and \(a\) is said to be von Neumann regular if there exists an element \(b\) in \(A\) such that \(Q(a)b\) and \(a\) coincide. The authors show that an element \(a\) in \(A\) is von Neumann regular if and only if the conjugate linear operator \(Q(a)\) has norm-closed range, and discuss the notion of von Neumann regularity in terms of the the spectral properties of elements of \(A\). The authors go on to define the quadratic co-norm \(\gamma^q(a)\) of an element \(a\) of \(A\) by \[ \gamma^a(a)= \text{inf}\{\| Q(a)(b)\|: b\in A,\,\text{dist}(b,\ker(Q(a)))\geq 1\}. \] It is shown that \(\gamma^q(a)\) coincides with the infimum of the squares of the elements in the triple spectrum of \(a\). This leads to the description of non-zero tripotents in \(A\) as those elements the norm and quadratic co-norm of which are both one. These results allow the authors to show that when \(A\) and \(B\) are \(\text{JBW}^*\)-triples, which are, of course, well-supplied with tripotents, a contractive linear bijection from \(A\) onto \(B\) is a triple isomorphism if and only if it preserves quadratic co-norms. In the final two sections of this interesting paper, the authors study the norm-continuity of the quadratic co-norm in a \(\text{JB}^*\)-triple \(A\) and then apply all their results to the case when \(A\) is a \(C^*\)-algebra, thereby producing some new findings in this area.
scientific article; zbMATH DE number 5278754
scientific article