On Completely Positive Maps

ABSTRACT

Completely positive maps is an important eld due to its signicance,

application and mathematics itself. While discussing the properties of the

positive maps, researchers have questioned whether the properties of the

positive maps also hold for completely positive maps. In chapter 1, we

have started with a C-algebra A, generated on A other C-algebras and

then investigated these forms of C-algebras. We investigated whether

the properties of A such as self-adjointedness and completeness under

norm still hold on the C-algebras generated on A. In chapter 2, the condition

for the positivity of the elements of these generated C-algebras

is given. This has been done by showing that their inner product with

elements from a Hilbert space is positive. A unital contraction is necessarily

positive. Conditions under which positive maps are completely

positive are discussed. In chapter 3, boundedness and complete boundedness

of these maps have been investigated. This, we have done by

showing that, indeed, whenever the operator system is a C-algebra, then

a positive map is bounded and completely bounded, if its norm is equal

to its complete bound which must be nite. All completely positive maps

are completely bounded, however the converse is not always true. This

has been shown by giving examples and counter examples. The results

of this study will pave way for construction of new C-algebras from the

known ones, which will be helpful in the development of the research on

positive maps on these generated C-algebras and may also be applied by

mathematicians in solving spectral problems.