Adjusting new approaches to Banach homology based on a homomorphism with counterexamples
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In this paper, we use group algebras and semigroup algebras to provide counterexamples that demonstrate the failure of \cite[Theorem 2.1, Proposition 3.2]{gho}. Additionally, we revisit and refine their results. Specifically, we establish that if $A$ is a $\varphi$-biprojective Banach algebra with a closed ideal $I$ such that $\overline{AI}=I$, then $A/I$ is $\widetilde{\varphi}$-biprojective. Lastly, we explore the connection between $\varphi$-approximate Helemskii biflatness and $\varphi$-pseudo amenability.
Abstract
In this paper, we use group algebras and semigroup algebras to provide counterexamples that demonstrate the failure of \cite[Theorem 2.1, Proposition 3.2]{gho}. Additionally, we revisit and refine their results. Specifically, we establish that if $A$ is a $\varphi$-biprojective Banach algebra with a closed ideal $I$ such that $\overline{AI}=I$, then $A/I$ is $\widetilde{\varphi}$-biprojective. Lastly, we explore the connection between $\varphi$-approximate Helemskii biflatness and $\varphi$-pseudo amenability.