Three automorphism theorems for triangular matrix algebras Original Research Article

Let Tn(A) be the algebra of upper triangular n × n matrices with entries from an associative k-algebra A, where k is a commutative ring. Recently several authors (Barker, Coelho, Jøndrup, Kezlan) have shown that if A is sufficiently well behaved, then every k-algebra automorphism of Tn(A) decomposes into a product of an inner automorphism and an automorphism defined by an automorphism of A. In this paper we find new sufficient conditions for A that guarantee such decompositions, and we give comparisons with previous results. As an intermediate step we show that a single automorphism admits such a decomposition if and only if it respects the subspace of strict upper triangular matrices. We also consider the case of infinite matrices.