Generalized projections in Banach algebras
Let \({\mathcal J}(A)\) be the set of all elements of the form \(a=h+ik\), where \(h\) and \(k\) are Hermitian elements in a complex unital Banach algebra \(A\) and \(i\) is the imaginary unit. The involution \(*\) on \({\mathcal J}(A)\) is defined by \((h+ik)^*=h-ik\). An element \(a\in{\mathcal J}(A)\) is a generalized projection if \(a^2=a^*\). The author states and proves some properties of generalized projections. He also finds some sufficient and some both sufficient and necessary conditions under which an element from \({\mathcal J}(A)\) is a generalized projection. In the last section, some of these results are generalized for the case of \(q\)-generalized projections for \(q\in\mathbb N\) and \(q>1\), i.e., for elements \(a\in{\mathcal J}(A)\) such that \(a^q=a^*\).
scientific article; zbMATH DE number 5382402