Non-commutative Schur-Horn theorems

MASSEY, PEDRO

Regina-Canada

Congreso; Canadian Operator Algebra symposium; 2009

University of Regina, Regina, Canada

Let A be a unital *-subalgebra of the algebra of all n by n complex matricesand let E denote the trace preserving conditional expectation onto A. If U(b)(resp. C(b)) denotes the unitary (resp. contractive) orbit of the positive matrix"b" then we give necessary and sucient conditions for the matrix a in A tobe in E(U(b)) (resp. E(C(b))); these conditions are related with the Horn-Klyachko´s equations. We consider these results as non-commutative extensionsof the classical Schur-Horn theorem. We use the previous results to introducewhat we call extended (sub)-majorization between positive matrices. We studysome Jensen´s-type inequalities associated with the extended majorization andapply these notions in the context of protocols (general linear reconstructionsystems).