Module Left Biflatness for Banach Algebras Related to Characters

Abstract

Amenability of Banach algebras is a fundamental concept in harmonic analysis. Over time, the notions of biflatness, $\varphi$-biflatness, and left $\varphi$-biflatness (where $\varphi$ is a character on the Banach algebra) were introduced and studied, such that each concept is weaker than the previous one, and various results clarifying the relationships among these concepts were obtained. In this paper, we extend the notion of left $\varphi$-biflatness to the module setting, which is a new and important concept, for the first time. Several new results for this generalized notion are established, and its connections with the earlier concepts are investigated.