Advances in Operator Theory and Inequalities

Representations induced by general positive sesquilinear maps with values in ordered Banach bimodules such as commutative and non-commutative
L1-spaces and the spaces of bounded linear operators from a
C-algebra into the dual of another C-algebra are considered. As a starting point, a generalized Cauchy-Schwarz inequality is proved for these maps and a representation of bounded positive maps from a (quasi) -algebra into such an ordered Banach bimodule is derived and some more inequalities for these maps are deduced. In particular, an extension of Paulsen’s modified Kadison-Schwarz inequality for 2-positive maps to the case of general positive maps from a unital -algebra into the space of trace-class operators on a separable Hilbert space and into the duals of von-Neumann algebras is obtained. Also, representations for completely positive maps with values in an ordered Banach bimodule and Cauchy-Schwarz inequality for infinite sums of such maps are provided. Concrete examples illustrate the results.