International Seminar Series on Advances in Operator Theory and Inequalities #2

In this talk, generalized Cauchy-Schwarz inequalities for positive sesquilinear maps with values in noncommutative
Lp-spaces for p > 1 are obtained. Bound estimates for their real and
imaginary parts are also provided, and, as an application, a generalization of the uncertainty relation in the context of noncommutative L2-spaces is given. Next, a Cauchy-Schwarz inequality
for positive sesquilinear maps with values in the space of bounded
linear operators from a von Neumann algebra into a C*-algebra
equipped with the numerical radius norm is proved. In the same
spirit, a new norm on a noncommutative L2-space, which generalizes the classical numerical radius norm of bounded linear operators
on a Hilbert space, is proposed, and a Cauchy-Schwarz inequality
for positive sesquilinear maps with values in the space of bounded
linear operators from a von-Neumann algebra into the noncommutative L2-space equipped with this new norm is proved. These results are used to get representations of general positive linear
maps with values in a non-commutative Lp-space and into cer
tain operator spaces in several different situations. Some concrete
examples are also given.