On completely bounded Operators

Abstract

Calculating norms of matrices when the entries are not constants is the first problem tackled in this thesis. We have considered the space of matrices with entries from the algebra of bounded linear operators and have managed to approximate norm in this space. The basic idea has been to identify this space with the space of bounded operators from H" (where 'H" is the orthogonal sum of n-copies of 7-{)to 'H'" and calculating the norm of an operator on H", This forms the content of chapter two. The notion of completely bounded operators is a fairly new and developing area in Mathematics. It started its life in the early 1980's following Stinespring's and Arveson's work on completely positive operators. It later gave rise to operator spaces, a new branch in operator algebra. Progress in this new area of Mathematics has been rapid and it is difficult to say which results motivated others. We have investigated the norm of completely bounded operators and have shown that they form an increasing sequence. The idea was to apply Hilbert-Schmidt norm to the definition of these operators. We have also given examples of these operators for illustration, something which is missing in the available literature. We have also investigated operator spaces, especially their algebraic tensor product. Specific interest has been in the matricial tensor product.