σ-Intertwinings, σ-Cocycles and Automatic Continuity

Hussien Mahdavian Rad, Assadollah Niknam

Abstract

Let $\mathcal{A}$ be  an  algebra, $\mathcal{X}$  an $\mathcal{A}$-bimodule and $\sigma: \mathcal{A} \to \mathcal{A}$ a continuous homomorphism.  In this paper, we show a continuous  linear one to one correspondence   between  $Z_{\sigma}^1 (\mathcal{A},  \mathcal{F}) $, the set of all module valued $\sigma$-derivations and  $LI_{\sigma}(\mathcal{A}, \mathcal{X})$,  the set of all left $\sigma$-intertwining mappings,  where $\mathcal{F}=B(\mathcal{A}_{+}, \mathcal{X})$ and that $B(\mathcal{A}_{+}, \mathcal{X})$ is a $\sigma(\mathcal{A})$-bimodule. A similar fact is proved  between  $Z_{\sigma}^n (\mathcal{A},  \mathcal{F}) $,  the set of all $n$-$\sigma$-cocycles,  and $LI_{\sigma}^n (\mathcal{A}, \mathcal{X})$, the set of all $\sigma$-intertwining mappings in the last variables.  Also there exists a linear homeomorphism between  $\mathfrak{Z}_{\sigma}^1 (\mathcal{A},  \mathcal{F}) $, the set of all continuous module valued $\sigma$-derivations,  and $B(\mathcal{A}, \mathcal{X})$. Moreove, it is proved that the same relation satisfies between  $\mathfrak{Z}_{\sigma}^n (\mathcal{A},  \mathcal{F}) $ and $B^n (\mathcal{A}, \mathcal{X})$.