# Amenable Banach algebras — homological characterization

Recall from Theorem VII.2.19 of Helemskii's monograph "The homology of Banach and Topological Algebras" that amenability of a Banach algebra $A$ is equivalent to any of the conditions below:

(i) derivations into dual modules are inner,

(ii) $\mathcal{H}_1(A,X)=0$ and $\mathcal{H}_0(A,X)$ is Hausdorff for all $A$-bimodules $X$.

The condition $\mathcal{H}_0(A,X)$ means that the image of the map

$d_0\colon A\widehat{\otimes}X\to X,\,\,\,\,d_0(a\otimes x):=a\cdot x-x\cdot a$

is closed. Now, when proving $(i)\Rightarrow(ii)$ the closedness of $\operatorname{im}d_0$ follows by a general fact (see Lemma 0.5.1 in this same Helemskii's monograph). My question is: is it possible to show closedness of $\operatorname{im}d_0$ directly?

The reason for a direct proof lies in the fact that I am working beyond Banach algebra category where -- in particular -- Open Mapping Theorem is not available.

• What category are you using? If I recall correctly, going from "$H^1(A,X^*)=0$" to "$H_0(A,X)$ is Hausdorff" uses duality theory of Banach spaces, but not the open mapping theorem. However, it has been a long time since I worked through the details from first principles – Yemon Choi Jul 12 '17 at 3:41
• If I guess correctly, you're thinking about the relation $X$ - flat $\Leftrightarrow$ $X^*$ - injective. In the category I am working in (DF-spaces) duals of DF are not DF. Therefore I turned my attention to Lemma 0.5.1. But this Lemma needs the OMT. Therefore my last chance'' is a direct proof. That's at least all I can figure out. – Krzysztof Jul 13 '17 at 8:33
• Does projecteuclid.org/euclid.hha/1251832561 help at all? – Yemon Choi Jul 13 '17 at 18:07
• I know this paper by Pirkovskii - it is mainly devoted to Fr\'echet algebras. The crucial fact (from the view point of my category) he is using is the duality between exact sequences, i.e. a short sequence of Fr\'echet spaces is exact iff its dual sequence is exact. Although DF-spaces are dual to Fr\'echet ones, the above fact is not true in this category. Of course I can take the bidual sequence to the initial one but a DF-space is not in general a subspace in its second dual. – Krzysztof Jul 18 '17 at 12:03