Chebyshev centers and approximation in pre-Hilbert C ∗ -modules

We extend the study of Chebyshev centers in pre-Hilbert C-modules by considering the C-algebra valued map defined by |x| = hx, xi1/2. We prove that if T is a remotal subset of a pre- Hilbert C-module M, and F  M is star-shaped at a relative Chebyshev center c of T with respect to F, then |x − qT (x)|2  |x−c|2 +|c−qT (c)|2(x 2 F). The uniqueness of Chebyshev center follows from this inequality. This is a generalization of a well-known result on Hilbert spaces