Banach bimodule-valued positive maps: inequalities and representations

In this paper we construct representations of general positive sesquilinear maps with values in ordered Banach bimodules such as commutative and non-commutative L 1 -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 positivemaps 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.