On invertibility preserving linear maps, simultaneous triangularization and Property L

1997/02/18 by Erik Christensen, Christensen, Erik
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #Operator Algebras (math.OA) #Rings, Modules, and Algebras #funct-an #math.OA

paper · pdf · doi:10.48550/arxiv.funct-an/9702017

amstex, 18 pages

arxiv created 1997/02/18 · openalex publication_date 1997/02/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Simultaneous similarity of matrices

Related

• Banach algebra mappings preserving the invertibility of linear pencils
• Linear maps preserving rank of tensor products of matrices
• Invertibility preserving mappings onto finite C*-algebras
• Some generalisations of the inequalities for positive linear maps
• (Real)linear preservers of multiples of unitaries and matrix pairs with some extremal norm properties