$σ$-Biderivations and $σ$-commuting maps of triangular algebras

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

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