Cracoviensis Studia Counter examples for pseudo-amenability of some semigroup algebras

. In this short note, we give some counter examples which show that [11, Proposition 3.5] is not true. As a consequence, the arguments in [11, Proposition 4.10] are not valid.


Amir Sahami
Let S be an inverse semigroup. The restricted product of x and y of S is xy if x * x = yy * and undefined otherwise. The set S with the restricted product forms a discrete groupoid. If we adjoin a zero element 0 to this groupoid and put 0 * = 0, we get an inverse semigroup denoted by S r . The multiplication of S r defined as follows The semigroup S r is called restricted semigroup see [2].
Example 1 Let S = N be the set of all natural numbers. Equip this set with "∨" operation, that is, m∨n = max{m, n} for all m, n ∈ S. Also equip S with "∧" operation, that is, m ∧ n = min{m, n} for all m, n ∈ S. With these two productions S becomes a commutative infinite inverse semigroup.
Example 2 Let S = G be a discrete group. It is well-known that S is an inverse semigroup.

Theorem 3 ([11, Proposition 3.5]) Let S be an inverse semigroup. Then 1 (S) is pseudo-amenable if and only if S is finite.
Using the previous theorem, the following result is given. We recall that a Banach algebra A is approximately amenable, if for each Banach A-bimodule X and each bounded derivation D : for more details see [9] and [10]. In the following examples we show that Theorem 3 and the proof of Theorem 4 are not valid.
Example 6 Let S = N. Equip this semigroup with ∧ and ∨ operations. We denote S by N ∧ and N ∨ , respectively. Using [5, Example 10.10], we see that 1 (N ∧ ) and 1 (N ∨ ) are approximately amenable. Clearly, δ 1 is an identity for 1 (N ∨ ). Also (δ n ) n∈N is a bounded approximate identity for 1 (N ∧ ). Applying Theorem 5 we get that 1 (N ∧ ) and 1 (N ∨ ) are pseudo-amenable semigroup algebras. Since N ∧ and N ∨ are inverse semigroups, by Theorem 3, N ∧ and N ∨ must be finite which is impossible.
Counter examples for pseudo-amenability of some semigroup algebras

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Example 7 Let S = G be a discrete and infinite amenable group. By the Johnson Theorem [12, Theorem 2.1.8], 1 (G) is amenable. So 1 (G) is pseudo-amenable. Since G is an inverse semigroup by Theorem 3, G must be finite, which is impossible.
It is shown in the proof of [11,Proposition 3.5] that every amenable inverse semigroup is finite which is not true. For instance, every bicyclic inverse semigroup is amenable but not finite and hence one side of the mentioned proposition is not true. In fact, it should be changed as follows: In light of the above discussion, [11,Propositon 4.10] can not be valid. In other words, when 1 (S) is pseudo-amenable, then by [11,Proposition 3.5], S is amenable (not finite) and in general case, amenability of S does not imply amenability or character amenability of 1 (S). This happen when, S is an abelian or left (right) cancellative unital semigroup. On the other hand, for module version of amenability, we have (ii) 1 (S) is module pseudo-amenable if and only if S is amenable [4].
(iii) 1 (S) is module character amenable if and only if S is amenable [3].