Characterization Of Norm Attaining Operators In C*-Algebras

Abstract

Various mathematicians have studied the subject of norm-attaining operators since its inception. Many forms of numerical ranges have been established which include the essential and the joint numerical ranges. However numerical ranges of norm-attaining operators in C*-algebras have not been fully investi- gated. Spectra of various operators have been studied and used to characterize other operators. However, spectra of norm attaining operators in C*-algebras are interesting and have not been fully investigated. Norms of various operators like elementary operators and others have also been studied over time by sev- eral authors and various results have been established. However in C*-algebras, norms of norm-attaining operators still remain interesting to mathematicians. This study focused on characterizing norm-attaining operators in C*-algebras. The specific objectives were to: characterize numerical ranges of norm attain- ing operators in C*-algebras, characterize spectra of norm-attaining operators in C*-algebras and establish norms of norm-attaining operators in C*-algebras. The methodology involved fundamental theorems like the Riesz Representation and Toeplitz-Hausdorff Theorems to characterize numerical range. In addition we employed inequalities such as Cauchy-Schwarz, triangle inequality and Po- larization identity in establishing the norms. The results obtained show that; the numerical range of a norm-attaining operator S is non-empty and is equal to the convex hull of its point spectrum. In addition, the spectra is bounded and closed. Lastly, kSo + S1 k = kSo k + kS1k is equivalent to kSo kkS1k ∈ W (S∗ S1) where So , S1 ∈ N A(H ). The results obtained are contributions of knowledge to C*-algebras and operator theory and a motivation to a further research. They may also be useful in mathematical formulation of quantum mechanics.