On invertibility preserving linear maps, simultaneous triangularization and Property L

TL;DR

This paper investigates conditions under which invertibility-preserving linear maps on matrix algebras are homomorphisms, and explores properties related to simultaneous triangularization and Property L in matrix theory.

Contribution

It establishes that invertibility-preserving maps are homomorphisms modulo the Jacobson radical under certain dimensional conditions and provides new results on Property L and simultaneous triangularization.

Findings

Invertibility-preserving maps are homomorphisms modulo the Jacobson radical under specified conditions.

Provides sufficient conditions for the simultaneous triangularization of matrix sets.

Contains results related to Property L in matrix algebras.

Abstract

Let F be a linear unital map of a unital matrix algebra A over the complex numbers into the complex n by n matrices. Then F induces a linear unital map Fk of the k by k matrices over A into the complex nk by nk matrices by the action of F on each entry in the k by k matrices. If Fk preserves invertibility and k is greater than dim(alg(F(A)))-dim(F(A))+2 then F is a homomorphism modulo the Jacobson radical in alg(F(A)). The paper also contains some results related to the property L and some sufficient conditions for simultaneous triangularization of sets of matrices.