BZ Seminar in Analysis: Eric Carlen (Rutgers)

Inequalities for Schwarz maps on matrix algebras

A linear map T from the space of n by n matrices to the space of m by m matrices is a Schwarz map in case T(A^*A) - T(A)^*T(A) is positive semidefinite. Evidently every such map T is positive, but not all positive maps are Schwarz maps. On the other hand, every quantum Markov map (completely positive with T(I) = I) is a Schwarz map but again, the inclusion is strict.

This talk will present joint work with Mueller-Hermes and Zhang in which duality is exploited to give simple proofs of known and new  inequalities for Schwarz maps. It will be shown how these yield celebrated convexity and concavity theorems, such as those of Lieb and Epstein as simple corollaries.  The ideas may be applied in a more elaborate operator algebra setting, but by presenting them in the matrix algebra setting, the talk will require no specialized knowledge as background. 

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