Orbits of positive operators from a differentiable viewpoint
Let \(\mathcal A\) be a unital \(C^*\)-algebra represented on a Hilbert space \(\mathcal H\), \(G\) be the group of invertible elements of \(\mathcal A\), and \({\mathcal A}^+_{cr}\) the subset of \({\mathcal A}^+\) of positive operators with closed range. In the present paper, the authors study the geometric structure of the orbits \({\mathfrak A}_a=\{gag^*\mid g\in G\}\) corresponding to elements of \({\mathcal A}^+_{cr}\). The authors provide the orbit \({\mathfrak A}_a\) with a structure of a differentiable homogeneous space such that each component belonging to \({\mathfrak A}_a\) remains a submanifold. The case \({\mathfrak A}=L(H)\), the algebra of all bounded linear operators on \(\mathcal H\), is studied in detail in order to show that some changes have to be made in the topology of \({\mathfrak A}_a\) to obtain a structure of that type.
scientific article; zbMATH DE number 2081975