Character injectivity and projectivity of Banach modules

Abstract

Let A be a Banach algebra and let φ be a non-zero character on A. We introduce and study the notions φ-injectivity and φ-projectivity for Banach A-modules ℰ by modifying the injectivity and projectivity of ℰ. We give some necessary and sufficient conditions for φ-injectivity and φ-projectivity of ℰ. In sequel, we establish an important relationship between φ-amenability of A and φ-injectivity of ℰ. This gives a positive answer to a question of Hellemskii in the setting of injectivity based on φ.