International Journal of Pure and Applied Mathematics Volume 99 No. 2 2015, 191-200 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v99i2.6 P A ijpam.eu FARTHEST POINTS IN HILBERT OPERATOR SPACES WITH APPLICATIONS M. Iranmanesh 1 § , F. Soleimany 2 1,2 Department of Mathematical Sciences Shahrood University of Technology P.O. Box 3619995161-316, Shahrood, IRAN Abstract: The purpose of this paper is to Provide conditions for the existence of farthest points of closed and bounded subsets of Hilbert operator spaces. This will done by applying the concept of numerical range. We give, inter alia, some results to characterize farthest points of a subset of a C ∗ -algebra A from a fixed element x ∈ A. Meanwhile, we point out the main theorems of R. Saravanan and R. Vijayaragavan[11] are incorrect, by given two counterexamples. AMS Subject Classification: 41A50, 41A52, 41A65, 46L05, 47A58 Key Words: farthest point, strong farthest point, numerical range, C ∗ - algebras 1. Introduction The problem of farthest points in normed linear spaces has been studied by Singer, Franchetti and T.D. Narang, in [4, 9]. They give some results on characterization and existence of farthest points in normed linear spaces in terms of bounded linear functionals. Also several results related with farthest points in the context of normed linear space and metric space can be obtained in[1, 3, 5, 6, 10, 11].Saravanan and R. Vijayaragavan [11] characterize farthest points from bounded sets with respect to uniform norm. We first (see Section 2) give some preliminary concepts and definitions on C ∗ -algebras. In Section 3 the existence of farthest points will be discussed. We also present the results Received: November 1, 2014 c  2015 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author