Norm of elementary operator in tensor product of c∗-algebras

Abstract

Many properties of elementary operators, including spectrum, numerical ranges, compactness, rank, and norm have been studied in depth with the norm property attracting many researchers due to its wide range of applications. Generally, the calculation of norms involves finding a formula that describes the norms in terms of their coefficients. The discussion on the norm of the elementary operator can be traced to Stampfli’s theorem of 1970 which created a proper base for the study of norms. The determination of the norm of the elementary operator in C∗-algebras, JB*-algebras, standard operator algebras, Cartan factor, prime C∗-algebras, two-dimensional complex Hilbert space and tensor product have been studied and based on earlier research, the norm of basic elementary operator in tensor product of C∗-algebras have been evaluated using the concept of finite rank operator and Stampfli’s maximal numerical range. The norms of other types of elementary operators in tensor product of C∗-algebras was not determined. This study extended the study of the norm of elementary operator in tensor product of C∗-algebras to the general finite length elementary operator. The study determined the norm of finite length elementary operator in tensor product of C∗-algebras and found for a finite length elementary operator, Tn, in a tensor product of C∗algebras . Consequently, the conditions under which the norm of an arbitrary finite length elementary operator in tensor products of C∗-algebras as is expressible in terms of the norms of its coefficient operators were also established and found that if ||Ai ⊗ Bi|| ∈ W◦(Ai ⊗ Bi) and ||Ci ⊗ Di|| ∈ W◦(Ci ⊗ Di) ∀ i = 1,2,...,n. Finally, the research determined the norm of Jordan elementary operator in the tensor products of C∗algebras where for every tensor product X ⊗ Y ∈ B(H ⊗ K) with ∥X ⊗ Y ∥ = 1, then ∥UA⊗B,C⊗D∥ = 2∥A∥∥B∥∥C∥∥D∥, A,C ∈ B(H), B,D ∈ B(K) and UA⊗B,C⊗D is the Jordan elementary operator in a tensor product of C∗-algebras. The techniques of rank one operator,finite rank operator, the definition of the elementary operator in tensor product, properties of tensor product, inner product, properties of functionals, and the norm were used to achieve the objectives of this study. The finding of this research can be used in areas of functional analysis, linear algebra, operator theory and mathematical physics by utilizing various properties of an elementary operator under the study of C∗-algebra.