Schur Complements in C∗− Algebras

The formula (1) was first used by Schur [22], but the idea of the Schur complement goes back to Sylvester (1851), and the term Schur complement was introduced by E. Haynsworth [16]. In the beginning Schur complements were used in the theory of matrices. M.G. Krein [19] and W.N. Anderson and G.E. Trapp [4] extended the notion of Schur complements of matrices to shorted operators in Hilbert space operators, and Trapp defined the generalized Schur complement by replacing the ordinary inverse with the generalized inverse. Schur complements and generalized Schur complements were studied by a number of authors, have applications in statistics, matrix theory, electrical network theory, discrete-time regulator problem, sophisticated techniques and some other fields (see [20], [11], [10], [5], [6]). In this paper we introduce and study the Schur complement of positive elements in a C∗-algebraA and among other things, we embark study the extremal characterizations of Schur complement. LetA be a complex C∗-algebra with the unit 1. The Moore-Penrose inverse of an element a ofA is the unique element a† of A satisfying the equations aa†a = a, a†aa† = a†, (aa†)∗ = aa†, (a†a)∗ = a†a (see [14], [15], [17], [21]). The set of all a ∈ A that possess the Moore-Penrose inverse will be denoted by A†. It is shown in ([14], [18]) that a ∈ A† if and only if a ∈ aAa. We also write A−1 for the set of all invertible elements inA. The word ‘projection’ will be reserved for an element q ofAwhich is self-adjoint and idempotent, that is, q∗ = q = q. In this paper Ah stands for the set of all selfadjoint elements of A. The symbols A•h, A•h and A+ denote the sets of all idempotent, projection and positive elements of A, respectively. If a, b ∈ Ah and a − b ∈ A+, we write a ≥ b (or b ≤ a). We say that a ∈ A is relatively regular, provided that there exists some b ∈ A such that aba = a. In this case b is called an inner generalized inverse of a. We use a− to denote an arbitrary inner generalized inverse of a.