Matrix systems, algebras, and open maps

TL;DR

This paper investigates the topological properties of state restrictions on matrix algebras and subalgebras, showing restrictions to subalgebras are open, which simplifies certain problems in matrix theory and quantum information.

Contribution

It proves that restrictions of states to *-subalgebras of matrix algebras are open, providing a new insight into the topology of state spaces in matrix theory.

Findings

Restrictions to *-subalgebras are open

Simplifies topology problems in matrix theory

Impacts quantum information theory

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.