## On norms of elementary operators

##### Abstract/Overview

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.