On Norm – Attainability of Elementary Operators in Tensor Product of C^* - Algebras

Abstract

The study of elementary operators has been extensively explored by many mathematicians due to its wide range of applications. The researchers have focused on aspects such as the norm, numerical ranges, spectrum, compactness, norm - attainability and orthogonality. For instance, determination of norm - attainability of elementary operators in Hilbert and Banach spaces has been established. Generally, using the property of norm - attainability of operators, an operator T ∈ B(H) is said to be norm - attainable if there exists a unit vector  such that ||T || = ||T||. Equivalently, norm - attainability of generalized finite operators in  -algebra has also been determined. However, the property of norm - attainability in tensor products of  -algebra largely remains unknown. This paper therefore extends the study of norm- attainability of operators in Hilbert, Banach and  -algebra spaces to tensor products of  -algebra. In particular, the study determines the norm - attainability of elementary operators in tensor products of  -algebra using the property of norm - attainability of operators. The techniques and the properties of the norm and those of the tensor products will also be of great significance in achieving the desired results of this study.