On norms of elementary operators

Abstract

The study of elementary operators has been of great interest to many mathematicians for the past two decades. Of special interest has been to determine the norms of these operators. The norm problem for elementary operators involves finding a formula which describes the norm of an elementary operator in terms of its coefficients. The upper estimates of these norms are easy to find but approximating these norms from below has proved to be difficult in generaL Several mathematicians have produced known results for special cases on the lower estimates, for example, Mathieu found that for prime C*-algebras, the coefficient is ~, Stacho and Zalar obtained 2( v'2-1) for standard operator algebras on Hilbert spaces, Cabrera and Rodriguez obtained 20!I2 for JB* -algebras while Timoney came up with a formula involving the tracial geometric mean to calculate the norm of elementary operators. An operator T: A ~ A is called an elementary operator if T can be expressed in the formZ'[z) = L~=Iaixbi, \j x E A where A is an algebra and tu, b; fixed in A. The norm of an operator T is defined by IITII= sup{IITxll : x E H, Ilxll = I} where H is a Hilbert space. The purpose of this study therefore, has been t,o determine the lower estimate of the norm of the basic elementary operator on a' C*- algebra through tensor products. To do this we needed to have a good background knowledge on functional analysis, general topology, operator theory and C*-algebras by understanding the existing theorems and relevant examples especially on tensor product norms. We used the approach of tensor products in solving our particular problem. We hope that the results obtained shall be useful to applied mathematicians and physicists especially in quantum mechariics.