Variational and convexity properties of families of involutions
Let \(\mathcal S\) be the set of all \(C^*\)-algebra structures that share a given underlying topological algebra \(\mathcal A\), which is called Star on \(\mathcal A\). The problem is to study natural geometric structures of the set \(\mathcal S\). The following are some results shown in the article. 1) The set \(\mathcal S\) has a natural structure of an analytic manifold. Let \(\text{Int}({\mathcal A})\) be the group of inner automorphisms of \(\mathcal A\). Then \[ \sigma\to L_\phi \sigma,\quad \phi\in \text{Int}({\mathcal A}) \] is an analytic map, where \(L_\phi \sigma:= \phi \sigma \phi^{- 1}\). Let \(M\) be the typical orbit of this action: \[ M= \{L_\phi \sigma;\;\phi\in \text{Int}({\mathcal A})\}, \] which is called a constellation in the sky of \(\mathcal A\). We also have \[ T_\sigma M\simeq \{\text{ad}(x); \text{ad}(x)\sigma= 0,\;x\in {\mathcal A}\}, \] where \(T_\sigma M\) is the tangent space at \(\sigma\in M\). The orbit \(M\) is equipped with a reductive homogeneous space structure and a Finsler metric. At the tangent space \(T_\sigma M\) the metric is defined by means of a fixed conditional expectation \(\Phi\) onto the center \({\mathcal Z}({\mathcal A})\). 2) A presentation of each orbit \(M\) as the base of a fibration of the space \(G^\sigma\) of \(\sigma\) positive invertible elements of \(\mathcal A\) is given, where \(\sigma\) is any element of \(M\). As a consequence, we get that (1) geodesics in \(M\) are curves of minimal length. (2) any pair of stars in \(M\) can be joined by a unique geodesic. 3) The geodesic distance function of \(M\) is a convex function, that is, for any fixed \(\sigma\in M\), the function \(f_\sigma(\theta):= \text{dist}(\theta, \sigma)\), \(\theta\in M\) satisfies an inequality \[ f_\sigma(\gamma(t))\leq (1- t) f_\sigma(\gamma(0))+ tf_\sigma(\gamma(1)), \] for each geodesic \(\gamma(t)\) in \(M\).
scientific article; zbMATH DE number 764057
scientific article