| dc.description.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. | en_US |
