In this paper, we present examples contrasting the two main results of the theory of ('; )-amenability which were introduced in [4]. In particular, we show that being ('; )-bi atness does not imply ('; )-approximate biprojectivity, and thus prove that [4, Theorem 3.4] is false. Our main example shows that for an in nite discrete amenable semigroup G, the Banach algebra `1(G) is (id; )-bi at for any 2 (`1(G)), but not (id; )-approximately biprojective. The reason for this is that we can prove that (id; )-approximately biprojective implies the left contractivity of `1(G), which in turn implies that G is nite- a contradiction. Using an essentially identical argument, we show that [4, Theorem 3.7], which connects ('; )-pseudo-amenability with ('; )- approximate biprojectivity, is also false. Our examples thus point to the need for more srtuctural hypotheses in the theory of gerneralized amenability.
