σ-Biderivations and σ-commuting maps of triangular algebras

Let $\A$ be an algebra and $\sigma$ an automorphism of $\A$. A linear map $d$ of $\A$ is called a $\sigma$-derivation of $\A$ if $d(xy) = d(x)y + \sigma(x)d(y)$, for all $x, y \in \A$. A bilinear map $D: \A \times \A \to \A$ is said to be a $\sigma$-biderivation of $\A$ if it is a $\sigma$-derivation in each component. An additive map $\Theta$ of $\A$ is $\sigma$-commuting if it satisfies $\Theta(x)x - \sigma(x)\Theta(x) = 0$, for all $x \in \A$. In this paper, we introduce the notions of inner and extremal $\sigma$-biderivations and of proper $\sigma$-commuting maps. One of our main results states that, under certain assumptions, every $\sigma$-biderivation of a triangular algebras is the sum of an extremal $\sigma$-biderivation and an inner $\sigma$-biderivation. Sufficient conditions are provided on a triangular algebra for all of its $\sigma$-biderivations (respectively, $\sigma$-commuting maps) to be inner (respectively, proper). A precise description of $\sigma$-commuting maps of triangular algebras is also given. A new class of automorphisms of triangular algebras is introduced and precisely described. We provide many classes of triangular algebras whose automorphisms can be precisely described.