\(\varphi\)-amenable and \(\varphi\)-biflat Banach algebras (Q459222)

Let \(\varphi\) be a continuous homomorphism on a Banach algebra \(A\). The authors investigate \(\varphi\)-amenability and study the notion of \(\varphi\)-biflatness. They prove that if \(\varphi\) is a continuous epimorphism on a \(\varphi\)-biflat Banach algebra \(A\) having a bounded approximate identity, then \(A\) is \(\varphi\)-amenable. They also prove that if \(A^{**}\) is \(\varphi^{**}\)-biflat, then \(A\) is \(\varphi\)-biflat.