Matrix systems, algebras, and open maps

Abstract

Every state on the algebra $M_n$ of complex nxn matrices restricts to a state on any matrix system. Whereas the restriction to a matrix system is generally not open, we prove that the restriction to every *-subalgebra of $M_n$ is open. This simplifies topology problems in matrix theory and quantum information theory.

27 pages; version 2: corrections in Section 9 (among others, reference [51] added); version 3: minor corrections in Section 9; comments welcomed