  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.
