# First and second cohomology groups of Banach algebras

Johnson in the introduction section (page 1) in "Cohomology in Banach algebras" ZBL0256.18014, wrote that Guichardet in [14,15] obtained for a Banach algebra $$A$$,

one has $$H^1(A,X)=H^2(A,X)=0$$, when $$X$$ is reflexive and $$xa=\phi(a)x$$ for some multiplicative linear functional $$\phi$$ and certain other conditions are satisfied.

I dont have access to Guichardet's notes. Can you give me another reference or the details of the above fact?

[14]: A. Guichardet, Sur l’homologie et la cohomologie des algèbres de Banach, C. R. Acad. Sci. Paris, Ser. A 262 (1966), 38–41; Zlb 0131.13101.

[15] A. Guichardet. Sur l’homologie et la cohomologie des groupes localement compacts. C. R. Acad. Sci. Paris Sér. A 262 (1966), 118–120

I use the below comment and see references. But this paper is written in French. Can you give me Proposition 1 and 2 in the first reference in English?

• Commented Aug 29, 2019 at 15:50
• It may be worth noting that these "certain other conditions" must be quite significant, because otherwise one could take $X$ to be one-dimensional, and then find many many many many many many examples of $A$ and $\phi$ where $H^1(A,{\mathbb C}_\phi)\neq 0$. Commented Aug 31, 2019 at 19:45

Some background context from [14] (the first of the links provided by François Ziegler):

Guichardet considers a Banach space $$H$$ equipped with a continuous representation $$\pi:A \to {\mathcal L}(H)$$ and a continuous anti-representation $$\rho:A \to {\mathcal L}(H)$$. He doesn't specify what notion of continuity he is using, but at this level of generality it is almost surely just norm-to-norm continuity of $$\pi$$ and $$\rho$$ as linear maps from $$A$$ to $${\mathcal L}(H)$$. He also states that when $$A$$ is unital then $$\pi$$ and $$\rho$$ will be assumed to be unital homomorphisms.

Guichardet then defines $$H^n(A,\pi,\rho)$$ to be what I would write as $$H^n(A, {}_\pi H_\rho)$$, i.e. the (bounded, continuous) Hochschild cohomology of $$A$$ with coefficients in the $$A$$-bimodule $${}_\pi H_\rho$$, with the notation indicating that the left action of $$A$$ on the Banach space $$H$$ is given by $$\pi$$ and the right action by $$\rho$$.

Proposition 1 in [14] says:

On suppose $$A$$ unitaire, $$H$$ réflexif, $$\rho$$ scalaire et $$\ker\rho$$ admettant une unité approchée; alors $$H^1(A,\pi,\rho)=0$$.

My (idiomatic) translation:

Suppose that $$A$$ is unital, $$H$$ is reflexive, $$\rho$$ is scalar-type and $$\ker\rho$$ has a bounded approximate identity. Then $$H^1(A,\pi,\rho)=0$$.

(notes:

1. scalar-type means: of the form $$a\mapsto \chi(a)I_H$$ for some homomorphism $$\chi:A\to{\mathbb C}$$. By Guichardet's earlier conventions, since $$A$$ is unital one should assume $$\chi(1_A)=1$$.

2. it is clear from the context, given the proof of Proposition 1, that Guichardet intends his "approximate identities" to be bounded)

Next, Proposition 2 in [14] says:

On reprend les hypotheses de la Proposition 1 et l'on suppose en outre que $$\ker\rho$$ contient une suite $$a_1,a_2,\dots$$ telle que l'idél gauche de $$\ker\rho$$ qu'elle engendre soit dense dans $$\ker\rho$$. Alors $$H^2(A,\pi,\rho)=0$$.

My (idiomatic) translation:

We keep the hypotheses of Proposition 1 and suppose in addition that $$\ker\rho$$ contains a sequence $$a_1, a_2,\dots$$ such that the left ideal generated by this sequence in $$\ker\rho$$ is dense in $$\ker\rho$$. Then $$H^2(A,\pi,\rho)=0$$.

There appear to be very few citations of these particular results. The paper [14] does not appear to have many citations and my impression is that those papers are merely paying tribute to Guichardet's definition of the cohomology groups, rather than using the results in Propositions 1 and 2.

Indeed, I believe that a much stronger result is true.

Theorem Let $$A$$ be a unital Banach algebra and let $$\phi:A \to {\mathbb C}$$ be multiplicative. Let $$X$$ be a Banach $$A$$-bimodule satisfying $$a\cdot x = \phi(a)x$$ for all $$x\in X$$ and $$a\in A$$.

If $$\ker\phi$$ has a bounded right approximate identity (i.e. a net $$(u_\alpha)$$ in $$\ker\phi$$ satisfying $$cu_\alpha \to c$$ for all $$c\in \ker\phi$$) then $$H^n(A,X^*)=0$$.

Notice that if $$H,\pi,\rho$$ satisfy the conditions of Proposition 1 in [14] then taking $$X=H^*=H_*$$ we see that the left action on $$X$$ satisfies the required condition in the Theorem, and $$\ker\phi=\ker\rho$$ satisfies the other required condition, so using Theorem 1 in the case $$n=1$$ we immediately obtain a proof of Proposition 1. Similarly, the assumption of Prop 2 are stronger than those in Theorem 1, and the conclusions of Theorem 1 are stronger than those in Prop 2, so Theorem 1 implies Prop 2 immediately.

Unfortunately I do not have a direct reference for this result, nor do I remember how exactly I learned it. The proof I have in mind comes from some old discussions with my PhD supervisor, who seemed to have picked it up by looking at proofs of less general results and working out the general case; he assumed everyone else had worked out for themselves, and so he never instructed me to find a specific reference.

My suspicion is that there exists a statement and proof of the result in the work of Helemskii or his (former) students — possibly Sheinberg? — but right now I don't have time to dig deeper into the literature. The reason for this belief is that there is a short conceptual proof of the theorem using homological methods (injective modules, flat modules, Ext functor, etc) which once again I learned from my PhD supervisor as a 2nd proof. So certainly one can derive this theorem quickly from results in Helemskii's book, but I don't remember if that book explicitly states this theorem itself. You can find an attempt to do things more concretely, in the case where $$A$$ is commutative, scattered throughout a paper I wrote in 2009 (but the parts of that paper which are most relevant to the Theorem above are also the least original).

By the way, it is for the reasons stated in the previous paragraph, that I believe most work on $$\phi$$-amenability or character amenability has been re-inventing the wheel. The Russian school knew many of these results in the 1970s/1980s.