On norms of a derivation

Abstract

The study of derivations still remains an area of interest to mathematicians today. Of special attention has been the study of norms of inner derivations. Most of the work in this area is based on Stampfli's result of 1970, where he established the equality between the norm of an inner derivation and twice the distance between an element of an algebra to the centre of that algebra, specifically for a primitive C*-algebra with an identity. This result has been extended by other mathematicians to other algebras, like Von Neumann, Calkin, W*· - algebras among others. In this study, we've continued to investigate Stampfli's result. In particular, we've used the approach of tensor product to establish the equality for the algebra of bounded linear operators on a Hilbert space. Further, we have explored the norm of inner derivations on norm ideals and established the relationships between norms of inner derivations restricted to algebras, norm ideals and the quotient algebra. On the other hand, an interesting relationship between the diameter of the numerical range and the norm of inner derivation has been established. Moreover, their applications to hyponormal and S - universal operators have been investigated. The methodology has been majorly based on the previous works of Stampfli, Fialkow, Kyle, Barraa and Boumazgour, among others. We also revisited related theories from operator algebra and analysis in general. In the operator - algebraic formulation of quantum theory, these results are useful to theoretical physicists and applied mathematicians alike. For pure mathematicians, we hope this will provide a motivation for further research in the development of the area.