Centralizing traces with automorphisms on triangular algebras

Abstract

Let ${\mathcal{T}}$ be a triangular algebra over a commutative ring ${\mathcal{R}}$, ${\xi}$ be an automorphism of ${\mathcal{T}}$ and ${\mathcal{Z}_{\xi}(\mathcal{T})}$ be the ${\xi}$-center of ${\mathcal{T}}$. Suppose that ${\mathfrak{q} \colon \mathcal{T} \times \mathcal{T} \longrightarrow \mathcal{T}}$ is an ${\mathcal{R}}$-bilinear mapping and that ${\mathfrak{T}_{\mathfrak{q}} \colon \mathcal{T} \longrightarrow \mathcal{T}}$ is a trace of ${\mathfrak{q}}$. The aim of this article is to describe the form of ${\mathfrak{T}_{\mathfrak{q}}}$ satisfying the commuting condition ${[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi}=0}$ (resp. the centralizing condition ${[\mathfrak{T}_{\mathfrak{q}}(x), x]_{\xi} \in \mathcal{Z}_\xi(\mathcal{T})}$) for all ${x\in \mathcal{T}}$. More precisely, we will consider the question of when ${\mathfrak{T}_{\mathfrak{q}}}$ satisfying the previous condition has the so-called proper form.