An Order-Preserving Linear Map from Matrices to Banach *-Algebras and Applications

Abstract

We establish a theorem for which a number of absolute-value identities and inequalities in the framework of Banach <svg style="vertical-align:-0.0pt;width:10.1px;" id="M1" height="7.9250002" version="1.1" viewBox="0 0 10.1 7.9250002" width="10.1" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.863)"><path id="x2217" d="M471 153q-22 -15 -61 -13q-25 31 -45.5 49.5t-56.5 41.5q4 -71 28 -134q-17 -33 -42 -46q-24 12 -42 46q24 63 28 134q-36 -23 -56.5 -41.5t-45.5 -49.5q-36 -2 -61 13q0 28 19 59q65 10 130 43q-65 33 -130 43q-19 31 -19 59q22 15 61 13q25 -31 45.5 -49.5t56.5 -41.5&#xA;q-4 71 -28 134q17 33 42 46q24 -12 42 -46q-24 -63 -28 -134q36 23 56.5 41.5t45.5 49.5q36 2 61 -13q0 -28 -19 -59q-65 -10 -130 -43q65 -33 130 -43q19 -31 19 -59z" /></g> </svg>-algebras can be generated. To do this job, we construct an order-preserving linear map from the vector space of 2-by-2 hermitian matrices to a hermitian Banach <svg style="vertical-align:-0.0pt;width:10.1px;" id="M2" height="7.9250002" version="1.1" viewBox="0 0 10.1 7.9250002" width="10.1" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.863)"><use xlink:href="#x2217"/></g> </svg>-algebras. This map can convert any suitable matrix ordering to a number of identities and inequalities in Banach <svg style="vertical-align:-0.0pt;width:10.1px;" id="M3" height="7.9250002" version="1.1" viewBox="0 0 10.1 7.9250002" width="10.1" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.863)"><use xlink:href="#x2217"/></g> </svg>-algebras. Hence, we obtain a number of analogues of the well-known results in a framework of hermitian Banach <svg style="vertical-align:-0.0pt;width:10.1px;" id="M4" height="7.9250002" version="1.1" viewBox="0 0 10.1 7.9250002" width="10.1" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.863)"><use xlink:href="#x2217"/></g> </svg>-algebras.