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On invertibility preserving linear maps, simultaneous triangularization and Property L

Erik Christensen
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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.

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Erik Christensen. 1997-02-18. On invertibility preserving linear maps, simultaneous triangularization and Property L. https://arxiv.org/abs/funct-an/9702017

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