THE FIRST COHOMOLOGY GROUP H^{1}(G, M)

ANTHONY J. NARKAWICZ (VIRGINIA TECH)

Abstract. This paper characterizes the first cohomology group H^{1}(G, M) where M is a Banach space (with norm || ||_{M}) that is also a left ℂG module such that the elements of G act on M as continuous ℂ-linear transformations. We study this group for G an infinite, finitely generated group. Of particular interest are the implications of the vanishing of the group H^{1}(G, M). The first result is that H^{1}(G, ℂG) imbeds in H^{1}(G, M) whenever ℂG ⊂ M ⊂ L^{p}(G) for some p ∈ ℕ. This is an unpublished result and shows immediately that if H^{1}(G, M) = 0, then G can have only 1 end. Secondly (also a new result), we show that H^{1}(G, M) is not Hausdorff if and only if there exist f_{i} ∈ M with norm 1 (||f_{i}||_{M} = 1) for all i with the property that ||gf_{i} − f_{i}||_{M} ⟶ 0 as i ⟶ ∞ for every g ∈ G. This is then used to show that if M and || ||_{M} satisfy certain properties and if G satisfies a “strong Følner condition,” then H^{1}(G, M) is not Hausdorff. For the second half of this paper, we give several applications of these last two theorems focusing on the group G = ℤ^{n}.

1. Introduction

Motivation for this paper comes from two papers, one by Mohammed E.B. Bekka and Alain Valette ([3]) and the other, an expository paper, by Edward G. Effros ([2]). The first paper examines the group H^{1}(G, L^{2}(G)) and focuses on the implications of the vanishing of this group. It shows the following:

(1) H^{1}(G, L^{2}(G)) is Hausdorff if and only if G is non-amenable.

(2) The G-module imbedding ℂG → L^{p}(G) induces an imbedding of H^{1}(G, ℂG) into H^{1}(G, L^{p}(G)), p ⩾ 1.

2000 Mathematics Subject Classification. Primary: 20J06; Secondary: 46L05, 43A15.

2 ANTHONY J. NARKAWICZ (VIRGINIA TECH)

(3) If H^{1}(G, L^{2}(G)) = 0, then G is non-amenable with just one end.

The first of these results is due in part to the following result by A. Guichardet ([1],Corollary 2.3 of Chapter III): H^{1}(G, L^{p}(G)) is not Hausdorff if and only if there exists a sequence e_{n} in L^{p}(G) such that ||e_{n}||_{p} = 1 for all n with the property that ||ge_{n} − e_{n}||_{p} ⟶ 0 for all g ∈ G. We show that L^{p}(G) (with norm || ||_{p}) may be replaced by any Banach space M (with norm || ||_{M}) that is a ℂG module and has the property that the elements of G act on M as continuous ℂ-linear transformations. This is not trivial from Guichardet’s theorem. In fact, the topology on H^{1}(G, L^{p}(G)) (induced by the L^{p}-norm topology on L^{p}(G)) is entirely different from the topology on H^{1}(G, M) which is induced by the norm topology on M. This can be used to show that if G satisfies a “strong Følner condition,” then H^{1}(G, M) is not Hausdorff.

As for the second result, we also show that when ℂG ⊂ M ⊂ L^{p}(G) for some p ∈ ℕ, the G-module imbedding ℂG → M induces an imbedding of groups H^{1}(G, ℂG) → H^{1}(G, M). This (along with a minor result in [3]) shows that if H^{1}(G, M) = 0, then G can have only one end.

The second paper (by E.G. Effros) characterizes the ℂ algebra C_{red}^{∗}(ℤ^{n}) and provides a very explicit description of this algebra. He shows first that C_{red}^{∗}(ℤ^{n}) ≅ C(𝕋^{n}) as ℂ algebras (where 𝕋^{n} is the n-torus). Showing that C_{red}^{∗}(ℤ^{n}) has no nontrivial idempotents of course then shows that neither does C(𝕋^{n}), which shows that 𝕋^{n} is connected. Though this result is completely trivial, it ensues some interesting mathematics and gives rise to the paper’s title: Why the Circle is Connected: An Introduction to Quantized Topology. What Effros’s paper does mainly for ours is give us an explicit isomorphism from C(𝕋^{n}) onto C_{red}^{∗}(ℤ^{n}).

Our first criterion for the Hausdorffness of H^{1}(G, M) (Theorem 3) and the isomorphism between C(𝕋^{n}) and C_{red}^{∗}(ℤ^{n}) allow us to show that for n ⩾ 1, H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) is not Hausdorff and so dim_{ℂ} H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = ∞.

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 3

Secondly, the isomorphism from C(𝕋^{n}) onto C_{red}^{∗}(ℤ^{n}) allows us to describe the groups H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) explicitly. This in turn shows that there is a natural sequence of imbeddings

0 → H^{1}(ℤ, C_{red}^{∗}(ℤ)) → H^{2}(ℤ^{2}, C_{red}^{∗}(ℤ^{2})) → H^{3}(ℤ^{3}, C_{red}^{∗}(ℤ^{3})) → …

This and the fact that H^{1}(ℤ, C_{red}^{∗}(ℤ)) is not Hausdorff allows us to show that for each n ⩾ 1, dim_{ℂ} H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = ∞.

Finally, our second criterion for the Hausdorffness of H^{1}(G, M) (Theorem 4) gives another (more simple) proof that H^{1}(ℤ, C_{red}^{∗}(ℤ)) is not Hausdorff.

I would like to give special thanks to my advisor, Dr. Peter Linnell. His accessibility and the opinions that he gave me about the problems I was working on were absolutely crucial to this paper.

2. Preliminaries and Definitions

Let G be an infinite, finitely generated group.

Definition 1. By a G module M, we will mean a ℂ vector space M along with a homomorphism of G into Aut(M).

Definition 2. G satisfies the strong Følner condition means that for every finite subset S of G and every ε > 0, there exists an N ∈ ℕ such that for every M ∈ ℕ there exists a finite subset X of G such that |X| > M and |X − g · X| < N for every g ∈ S.

Because this definition may seem tedious, we should note that if G satisfies this condition, then G must be amenable. Also, a simple calculation shows that ℤ satisfies the strong Følner condition.

Definition 3. ℂG is the free ℂ vector space with basis G. In other words, ℂG is the set of all finite formal sums of the form ∑_{g} a_{g}g where the a_{g} are in ℂ and the g are in G.

ℂG is a G module in the obvious way (G acts on an element of ℂG by multiplication).

Definition 4. For 1 ⩽ p < ∞, L^{p}(G) is the set of all formal sums (not necessarily finite) of the form ∑_{g} a_{g}g with the property that ∑_{g} |a_{g}|^{p} < ∞.

Definition 5. The norm || ||_{p} on L^{p}(G) is given by

|| ∑_{g} a_{g}g||_{p} = [∑_{g} |a_{g}|^{p}]^{1/p}.

L^{p}(G) is complete in the topology induced by the norm || ||_{p}, so L^{p}(G) is always a Banach space. There is a multiplication defined on ℂG in the natural way (a_{i}g_{i} · a_{j}g_{j} = (a_{i}a_{j})(g_{i}g_{j})). While ℂG is a ring (under componentwise addition and this multiplication as its ring multiplication), L^{2}(G) is not necessarily a ring. One can have α, β ∈ L^{2}(G) with α · β ∉ L^{2}(G). However, it is well known that if α ∈ ℂG and β ∈ L^{2}(G), then α · β ∈ L^{2}(G). L^{2}(G) is certainly a Banach space over ℂ and via this multiplication, ℂG may be considered a subset of B(L^{2}(G)) (the bounded linear operators on L^{2}(G)).

Definition 6. The operator norm of an element α ∈ ℂG is given by

||α||_{op} = Sup{||α · β||_{2} : β ∈ L^{2}(G), ||β||_{2} = 1}.

Definition 7. C_{red}^{∗}(G) is the metric space completion of ℂG under the operator norm || ||_{op}.

We immediately have that C_{red}^{∗}(G) is composed entirely of bounded linear operators on L^{2}(G) and that C_{red}^{∗}(G) is complete and therefore a Banach space. In addition, ℂG ⊂ C_{red}^{∗}(G) ⊂ L^{2}(G). We also know that G acts on C_{red}^{∗}(G) (in the obvious way) as continuous ℂ-linear transformations.

Now we turn our attention to the groups H^{1}(G, M). We will view these groups in two different (though of course equivalent) ways.

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 5

First, let A_{G} be the set of set maps f: G → M with the property that for all a and b in G, a · f(b) − f(ab) + f(a) = 0. Let B_{G} be the set of all such maps given by f(b) = b · α − α for some fixed α ∈ M. Then we have the following first definition of H^{1}(G, M).

Definition 8. H^{1}(G, M) is the quotient group A_{G}/B_{G}.

Now consider the ring ℤG of all finite formal sums of the form ∑_{g} z_{g}g with z_{g} ∈ ℤ for all g ∈ G. Recall that for an arbitrary ring R, an R module is projective if and only if it is a direct summand of a free R module (there are several definitions). We say that an infinite exact sequence of ℤG modules ⋯ → P_{1} → P_{0} is a projective resolution of ℤ as ℤG modules if it extends to an exact sequence ⋯ → P_{1} → P_{0} → ℤ → 0 of ℤG modules. The ℤG module structure on ℤ is given by g · z_{i} = z_{i} for g ∈ G, z_{i} ∈ ℤ. Let … →^{d_{2}} P_{1} →^{d_{1}} P_{0} →^{d_{0}} ℤ → 0 be an exact sequence of ℤG modules with ⋯ → P_{1} → P_{0} a projective resolution of ℤ. This first sequence induces another sequence

0 → Hom_{ℤG}(ℤ, M) →^{d_{0}^{∗}} Hom_{ℤG}(P_{0}, M) →^{d_{1}^{∗}} Hom_{ℤG}(P_{1}, M) →^{d_{2}^{∗}} …

which gives us our second definition of H^{1}(G, M).

Definition 9. H^{1}(G, M) = ker(d_{1}^{∗})/ im(d_{0}^{∗}).

As it turns out, this definition is independent of the choice of projective resolution … ⟶ P_{1} ⟶ P_{0} of ℤ as ℤG modules. For the final part of our paper, we will also need the following definition of H^{n}(G, M).

Definition 10. For n ⩾ 1, H^{n}(G, M) = ker(d_{n}^{∗})/ im(d_{n−1}^{∗}).

Note that the two definitions of H^{1}(G, M) are equivalent.

To define the topology on H^{1}(G, M), we employ the first definition of this group, A_{G}/B_{G}. The topology is induced by the topology of point-wise convergence on A_{G}. That is for f_{n} ∈ A_{G}, f_{n} ⟶ 0 means that f_{n}(g) ∈ M converges to 0 in the norm

6 ANTHONY J. NARKAWICZ (VIRGINIA TECH)

|| ||_{M} on M for every g ∈ G. We should note that this is where the topologies on H^{1}(G, L^{2}(G)) and on H^{1}(G, M) differ. The basic open sets in A_{G} are the f ∈ A_{G} such that ||f(g_{1})−a_{1}||_{M}, …, ||f(g_{n})−a_{n}||_{M} < ε for some choice of fixed g_{i} ∈ G, a_{i} ∈ M, ε > 0 and n ∈ ℕ. In other words, the set of all such sets forms a basis for the topology on A_{G}.

Finally, we will need to view C(𝕋^{n}) as a subset of C(𝕋^{n+1}) (especially in the proof of Theorem 8). We do this as follows. For f ∈ C(𝕋^{n}) and (z_{1}, …, z_{n+1}) ∈ 𝕋^{n+1}, define f(z_{1}, …, z_{n+1}) = f(z_{1}, …, z_{n}).

3. The group H^{1}(G, M)

Keep supposing that G is infinite and finitely generated. In addition, suppose that M is a Banach space with norm || ||_{M} that is a left ℂG module and satisfies the property that G acts on M as continuous ℂ-linear transformations. Note that this implies that M is a G module.

Theorem 1. Suppose that ℂG ⊂ M ⊂ L^{p}(G) for some p ∈ ℕ. Then the G-module imbedding ℂG → M induces an imbedding of groups H^{1}(G, ℂG) → H^{1}(G, M).

Proof: Our result stated above has not been published to date and could prove to be useful. We will, however, follow the proof of Z. Q. Chen as described in [3] (Proposition 1), which shows that the imbedding ℂG → L^{2}(G) induces an imbedding H^{1}(G, ℂG) → H^{1}(G, L^{2}(G)). Since G is finitely generated, suppose that S is a finite generating set for G. For an arbitrary G-module N, we define C^{n}(G, N) to be the set of all set maps from G^{n} to N. In the case n = 0, we set C^{0}(G, N) = N. We have maps

d_{0} : C^{0}(G, ℂG) → C^{1}(G, ℂG), δ_{0} : C^{0}(G, M) → C^{1}(G, M) and
d_{1} : C^{1}(G, ℂG) → C^{2}(G, ℂG), δ_{1} : C^{1}(G, M) → C^{2}(G, M)

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 7

defined by [d_{0}(a)](g) = g·a−a, a ∈ ℂG, [δ_{0}(f)](g) = g·f −f, f ∈ M, [d_{1}(h)](g_{1}, g_{2}) = g_{1}h(g_{2}) − h(g_{1}g_{2}) + h(g_{1}), [δ_{1}(f)](g_{1}, g_{2}) = g_{1}f(g_{2}) − f(g_{1}g_{2}) + f(g_{1}). Then

H^{1}(G, ℂG) = \frac{ker(d_{1})}{im(d_{0})} and H^{1}(G, M) = \frac{ker(δ_{1})}{im(δ_{0})}.

The proposed imbedding is of course a + (im(d_{0})) ⟼ a + (im(δ_{0})), so define the natural homomorphism

θ: ker(d_{1}) → \frac{ker(δ_{1})}{im(δ_{0})}

via a ⟼ a + (im(δ_{0})). Since ℂG ⊂ M, we have that im(d_{0}) ⊂ im(δ_{0}), so im(d_{0}) ⊂ ker(θ).

Let b ∈ ker(θ). Then im(b) ⊂ ℂG and b ∈ im(δ_{0}). So there exists f ∈ M such that for all g ∈ G, b(g) = g · f − f ∈ ℂG. We aim to show that f must lie in ℂG and thus b ∈ im(d_{0}). Thus, we want to show that f has finite support. Suppose f = ∑ a_{g}g. Note that for all h ∈ G, h ∑ a_{g}g − ∑ a_{g}g = ∑(a_{g} − a_{hg})hg ∈ ℂG. Thus, for for all h ∈ G, φ(h) = {g ∈ G : a_{g} − a_{hg} ≠ 0} is finite. Since S is a finite set, it follows that

F (G) = ⋃_{s∈S} φ(s)

is finite as well. We may assume that for all s ∈ S, s^{−1} ∈ S. Let X be the Cayley graph of G with vertex set G and edge set {(g, sg) : s ∈ S}. By assuming that S is closed under inverses, it follows that X can be viewed as an undirected graph. Thus, if r and q (elements of G) are connected by an edge, it follows that r = sq for some s ∈ S. Now consider the graph X − F (G). F (G) is finite, so there are finitely many connected components of X − F (G). Then there exists a component of X − F (G) that is infinite (since G is infinite). Let q and r be in this connected component. So there exist s_{1}, …, s_{n} ∈ S such that s_{1} ⋯ s_{n}q = r, and this path cannot pass through F (G), so we have the property that for all i, s_{i} ⋯ s_{n}q ∉ φ(s) for any s ∈ S. Since s_{i−1} ∈ S for every i, it follows that a_{s_{i−1}s_{i}···s_{n}q} = a_{s_{i}···s_{n}q} for every i and so a_{q} = a_{s_{n}q} = a_{s_{n−1}s_{n}q} = ⋯ = a_{r}. Thus, all of the a_{g}’s are equal for all the g’s in X − F (G). By virtue of there being infinitely g’s in X − F (G) and since

8 ANTHONY J. NARKAWICZ (VIRGINIA TECH)

f = ∑_{g} a_{g}g ∈ M satisfies ∑_{g} |a_{g}|^{p} < ∞, it follows that for all g in this connected component of X − F (G), a_{g} = 0. Thus, if a_{t} ≠ 0, it follows that t lies in one of the finite connected components of this graph, and there are only finitely many such components (Since F (G) is finite), so there are only finitely many such t. i.e. a_{i} = 0 for all but finitely many i, and thus f ∈ ℂG. □

Lemma 1. dim_{ℂ} H^{1}(G, ℂG) = |b(G)| − 1 where |b(G)| is the number of ends of G.

Proof: For a proof of this, see [3](Lemma 2). □

Theorem 2. Suppose that ℂG ⊂ M ⊂ L^{p}(G) for some p ∈ ℕ. If H^{1}(G, M) = 0, then G has exactly 1 end.

Proof: If H^{1}(G, M) = 0, then by Theorem 1, H^{1}(G, ℂG) = 0. The previous lemma then tells us that |b(G)| = 1 (G has exactly 1 end). □

Theorem 3. H^{1}(G, M) is not Hausdorff if and only if there exists f_{i} ∈ M with norm 1 (||f_{i}||_{M} = 1) for all i with the property that ||gf_{i} − f_{i}||_{M} ⟶ 0 as i ⟶ ∞ for every g ∈ G.

Proof: For this proof, we think of H^{1}(G, M) as the set A_{G} of set maps f from G to M satisfying a · f(b) − f(ab) + f(a) = 0 for all a, b ∈ G modulo the set B_{G} of maps h of the form h(g) = g · e − e for some e ∈ M. Saying that H^{1}(G, M) is Hausdorff is equivalent to saying that 0 is closed in H^{1}(G, M) which is of course equivalent to saying that B_{G} is closed in A_{G}. This is equivalent to saying that B_{G} is complete and is thus a Frechet space. Remember that the topology on A_{G} is that of point-wise convergence, that is f_{n}: G → M tends to zero if and only if f_{n}(g) ⟶ 0 for every g ∈ G in the norm || ||_{M} on M.

We have a continuous one-to-one map h: M → A_{G} whose image is B_{G} (the obvious map). Since we are no longer talking about a topology on B_{G} induced by || ||_{p}, saying that this map is continuous needs justification. Suppose that (e_{n}) ∈ M satisfies

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 9

||e_{n}||_{M} ⟶ 0. Fix g ∈ G. Then ||[h(e_{n})](g)||_{M} = ||ge_{n}−e_{n}||_{M} ⩽ ||g−1||_{M}·||e_{n}||_{M} ⟶ 0 as n ⟶ ∞, since ||g−1||_{M} is fixed. Since the topology on B_{G} is that of point-wise convergence, it follows that h(e_{n}) ⟶ 0, so h is continuous.

Now M is a Banach space, so M is certainly complete and a Frechet space. Since a continuous, bijective map between Frechet spaces has a continuous inverse, saying that B_{G} is a Frechet space is equivalent to saying that the inverse map from B_{G} to M is continuous. We claim that this is equivalent to saying that there does not exist a sequence e_{n} ∈ M such that such that ||e_{n}||_{M} = 1 for all n and ||ge_{n} − e_{n}||_{M} ⟶ 0 for all g ∈ G.

The inverse map from B_{G} to M is given by the following. Say f ∈ B. Then f is given by f(g) = g · e − e for some e ∈ M. This inverse map sends f to e. Suppose that there exists such a sequence e_{n} ∈ M with ||e_{n}||_{M} = 1 and ||ge_{n} − e_{n}||_{M} ⟶ 0 for all g ∈ G. Thus, the maps in B_{G} determined by e_{n} converge to 0 (point-wise). However, their image under this inverse map does not, which means that this map cannot be continuous.

Conversely, suppose that this map is NOT continuous. Noting that the topology on B_{G} is that of point-wise convergence, this means that there exists a sequence a_{n}(g) = ge_{n} − e_{n} ∈ B_{G} such that ||a_{n}(g)||_{M} ⟶ 0 for all g ∈ G, but e_{n} does not converge to 0 in M. Then there exists ε > 0 such that for all N ⩾ 1, there exists m ⩾ N such that ||e_{m}||_{M} ⩾ ε. Multiply e_{n} by 1/ε and call the new sequence e_{n}. Note that since ε > 0 is fixed, the sequence of pre-images of the new e_{n} under this map still converge to 0 for each g ∈ G. Now we can choose a subsequence b_{n} of e_{n} such that ||b_{n}||_{M} ⩾ 1 for all n. For each m, multiply b_{m} by the unique real number 0 < y_{m} ⩽ 1 such that the new element of M (rename it X_{m}) has norm 1. Since y_{m} ⩽ 1 for all m, it follows that for all m, ||X_{m}||_{M} = 1 and ||gX_{m} − X_{m}||_{M} ⩽ ||gb_{m} − b_{m}||_{M} ⟶ 0 for all g ∈ G. □

Theorem 4. Keep the same assumptions as above about G and M and suppose that ℂG ⊂ M ⊂ L^{p}(G) for some p ⩾ 2. Suppose further that for every α ∈ ℂG, ||α||_{1} ⩾ ||α||_{M} ⩾ ||α||_{p}. If G satisfies the strong Følner condition, then H^{1}(G, M) is not Hausdorff.

Proof: Since G is finitely generated, we may let G = {g_{1}, g_{2}, … }. For each n ∈ ℕ, let G_{n} = {g_{1}, …, g_{n}}. Because G satisfies the strong Følner condition, for each n ∈ ℕ, we may pick N_{n} ∈ ℕ such that for every M ∈ ℕ, there exists a finite subset X of G such that |X| > M and |X − g_{i} · X| < N_{n} for every g_{i} ∈ S_{n}. Given any such n, choose a finite subset X_{n} of G such that |X_{n}| > (n · N_{n})^{p} and |X_{n} − g_{i} · X_{n}| < N_{n} for every g_{i} ∈ S_{n}. Let

β_{n} = \frac{∑_{x∈X_{n}} x}{|| ∑_{x∈X_{n}} x||_{M}} .

Note that for each n, ||β_{n}||_{M} = 1. Fix any g_{i} ∈ G. Then for n ⩾ i, we have

||g · β_{n} − β_{n}||_{M} = \frac{||g ∑_{x∈X_{n}} x − ∑_{x∈X_{n}} x||_{M}}{|| ∑_{x∈X_{n}} x||_{M}} ⩽

\frac{||g ∑_{x∈X_{n}} x − ∑_{x∈X_{n}} x||_{M}}{|| ∑_{x∈X_{n}} x||_{p}} ⩽ \frac{||g ∑_{x∈X_{n}} x − ∑_{x∈X_{n}} x||_{1}}{|| ∑_{x∈X_{n}} x||_{p}} =

= \frac{2 · |X_{n} − g_{i} · X_{n}|}{|| ∑_{x∈X_{n}} x||_{p}} < \frac{2 · N_{n}}{|| ∑_{x∈X_{n}} x||_{p}} = \frac{2 · N_{n}}{(|X_{n}|)^{\frac{1}{p}}} < \frac{2 · N_{n}}{((n · N_{n})^{p})^{\frac{1}{p}}} = \frac{2}{n} → 0.

Then ||β_{n}||_{M} = 1 for all n and ||g · β_{n} − β_{n}||_{M} → 0 for every g ∈ G. By the previous theorem, H^{1}(G, M) is not Hausdorff. □

Corollary 1. Suppose that G satisfies the strong Følner condition. Then H^{1}(G, C_{red}^{∗}(G)) is not Hausdorff and dim_{ℂ}(H^{1}(G, M)) = ∞. □

4. An Application: What Do the Groups H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) and H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) look like to ℂ?

Let S(𝕋^{n}) be the square integrable functions on 𝕋^{n}. In other words, S(𝕋^{n}) is the set of functions f: T^{n} → ℂ such that the following integral exists and is finite:

∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |f(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1}.

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 11

Effros’s paper ([2]) shows us that we have isomorphisms θ_{n}: S(𝕋^{n}) → L^{2}(ℤ^{n}) and ϕ_{n}: C(𝕋^{n}) → C_{red}^{∗}(ℤ^{n}) both given by

f ⟼ ∑_{m_{1},...,m_{n}} a_{m_{1},...,m_{n}}Z_{1}^{m_{1}} ⋯ Z_{n}^{m_{n}} where

a_{m_{1},...,m_{n}} = ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} Z_{1}^{−1−m_{1}} ⋯ Z_{n}^{−1−m_{n}} · f(z_{1}, …, z_{n}) · \frac{1}{(2πi)^{n}} · dz_{n} ⋯ dz_{1}

For f ∈ S(𝕋^{n}), Effros’s paper ([2]) shows us that we also have the following.

||θ_{n}(f)||_{2} = [\frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |f(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1}]^{1/2}.

Lemma 2. For f ∈ C(𝕋^{n}), ||ϕ_{n}(f)||_{op} = sup_{z∈𝕋^{n}} |f(z)|.

Proof: Let d be the usual distance metric on ℂ^{n}. First, suppose that g ∈ S(𝕋^{n}) satisfies ||θ_{n}(g)||_{2} = 1.

⟹ ||ϕ_{n}(f)θ_{n}(g)||_{2} =

[\frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |g(z_{1}, …, z_{n})|^{2}|f(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1}]^{1/2} =

⩽ sup_{z∈𝕋^{n}} |f(z)| · [\frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |g(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1}]^{1/2} =

= sup_{z∈𝕋^{n}} |f(z)|.

⟹ ||ϕ_{n}(f)||_{op} ⩽ sup_{z∈𝕋^{n}} |f(z)|.

Now, without loss of generality, ||ϕ_{n}(f)||_{op} = 1. Suppose that sup_{z∈𝕋^{n}} |f(z)| > 1. So there is an ε > 0 such that sup_{z∈𝕋^{n}} |f(z)| > 1 + ε. Then there is a w ∈ 𝕋^{n} and 1 > δ > 0 such that |f(z)| > 1 + ε for d(z, w) ⩽ δ where d is the usual distance metric on ℂ^{n} (This is due to the continuity of f(z)). By ∫ ⋯ ∫_{d(z,w)⩽δ}, we will mean ∫ ⋯ ∫_{{d(z,w)⩽δ}∩𝕋^{n}}, the integral over the intersection of the δ disk around w (in ℂ^{n}) with 𝕋^{n}. Since

[\frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} ·1 · dz_{n} ⋯ dz_{1}]^{1/2} = 1,

12 ANTHONY J. NARKAWICZ (VIRGINIA TECH)

and since δ < 1, {z ∈ 𝕋^{n} : d(z, w) ⩽ δ} is properly contained in 𝕋^{n}. Thus, there exists a k ∈ ℝ^{+} such that k > 1 and a g ∈ S(𝕋^{n}) defined by g(z) = 0 for d(z, w) > δ and g(z) = k for d(z, w) ⩽ δ with the property that

||θ(g)||_{2}^{2} = \frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |g(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1} = 1.

Also, by the definition of g,

∫ ⋯ ∫_{d(z,w)⩽δ} |g(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1} =
= ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |g(z_{1}, …, z_{n})|^{2}dz_{n} ⋯ dz_{1}.

Thus,

1 = ||ϕ_{n}(f)||_{op} ⩾ ||θ_{n}(g)ϕ_{n}(f)||_{2} =

= [\frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |g|^{2}|f|^{2}dz_{n} ⋯ dz_{1}]^{1/2} =

= [\frac{1}{(2π)^{n}} ∫ ⋯ ∫_{d(z,w)⩽δ} |f|^{2}|g|^{2}dz_{n} ⋯ dz_{1}]^{1/2} =

⩾ [\frac{1}{(2π)^{n}} ∫ ⋯ ∫_{d(z,w)⩽δ}(1 + ε)^{2}|g|^{2}dz_{n} ⋯ dz_{1}]^{1/2} =

= (1 + ε)[\frac{1}{(2π)^{n}} ∫ ⋯ ∫_{d(z,w)⩽δ} |g|^{2}dz_{n} ⋯ dz_{1}]^{1/2} =

= (1 + ε)[\frac{1}{(2π)^{n}} ∫_{|z_{1}|=1} ⋯ ∫_{|z_{n}|=1} |g|^{2}dz_{n} ⋯ dz_{1}]^{1/2} = 1 + ε > 1,

A contradiction. Thus, ||ϕ_{n}(f)||_{op} = sup_{z∈𝕋^{n}} |f(z)|. □

Theorem 5. Let n ⩾ 1. Then H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) is not Hausdorff and so dim_{ℂ} H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = ∞.

Proof: Again, let d be the usual distance metric on ℂ^{n}. Suppose that H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) is Hausdorff. By Theorem 3, there cannot exist (f_{i}) ∈ C_{red}^{∗}(ℤ^{n}) = C(𝕋^{n}) such that ||ϕ_{n}(f_{i})||_{op} = 1 for all i with the property that ||ϕ_{n}(gf_{i} − f_{i})||_{op} ⟶ 0 as i ⟶ ∞ for every g ∈ ℤ^{n}. We will exhibit such an (f_{i}).

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 13

Since given an i, {(z_{1}, …, z_{n}) ∈ 𝕋^{n} : d((z_{1}, …, z_{n}), (1, …, 1)) > 1/i} and {(z_{1}, …, z_{n}) : d((z_{1}, …, z_{n}), (1, …, 1)) < 1/(2i) for all j} are disjoint open subsets of 𝕋^{n} with positive distance between them (this distance is 1/(2i)), for each i we have a function f_{i} ∈ C(𝕋^{n}) with image contained in the closed unit disk that is identically 0 on the first set and identically 1 on the second. By the previous lemma, ||ϕ_{n}(f_{i})||_{op} = sup_{z∈𝕋^{n}} |f_{i}(z)| = 1 for all i. Choose (m_{1}, …, m_{n}) ∈ ℤ^{n}. Let B_{i} = {(z_{1}, …, z_{n}) ∈ 𝕋^{n} : d((z_{1}, …, z_{n}), (1, …, 1)) ⩽ 1/i} Then since each f_{i} is zero on the set A_{i} = {(z_{1}, …, z_{n}) ∈ 𝕋^{n} : d((z_{1}, …, z_{n}), (1, …, 1)) > 1/i},

||ϕ_{n}(Z_{1}^{m_{1}} ⋯ Z_{n}^{m_{n}}f_{i}(z_{1}, …, z_{n}) − f_{i}(z_{1}, …, z_{n}))||_{op} =

= sup_{(z_{1},...,z_{n})∈𝕋^{n}} |Z_{1}^{m_{1}} ⋯ Z_{n}^{m_{n}}f_{i}(z_{1}, …, z_{n}) − f_{i}(z_{1}, …, z_{n})| =

= sup_{B_{i}} |Z_{1}^{m_{1}} ⋯ Z_{n}^{m_{n}}f_{i}(z_{1}, …, z_{n}) − f_{i}(z_{1}, …, z_{n})| ⩽

⩽ sup_{B_{i}} |Z_{1}^{m_{1}} ⋯ Z_{n}^{m_{n}} − 1| · Sup_{B_{i}}|f_{i}| =

= sup_{B_{i}} |Z_{1}^{m_{1}} ⋯ Z_{n}^{m_{n}} − 1| ⟶ 0 as i ⟶ ∞.

This is due to the fact that on B_{i}, d((Z_{1}, …, Z_{n}), (1, …, 1)) ⩽ 1/i and the fact that the m_{1}, ..., m_{n} are fixed. In other words, ||gf_{i} − f_{i}||_{op} ⟶ 0 as i ⟶ ∞ for every g ∈ ℤ^{n}. □

Now we turn our attention to the groups H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})). We view 𝕋^{n} as {(z_{1}, …, z_{n}) : |z_{i}| = 1 for all i}. Again from Effros’s paper ([2]), we know that for all n, C_{red}^{∗}(ℤ^{n}) ≅ C(𝕋^{n}) as ℤ^{n} modules. But what is the ℤ^{n} module structure on C(𝕋^{n})? We view ℤ^{n} as the free abelian group on the generators x_{1}, …, x_{n}. Then the action of an element x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}} on an element f(z_{1}, …, z_{n}) of C(𝕋^{n}) is given trivially by (x_{1}^{e_{1}} ⋯ x_{n}^{e_{n}}) · f(z_{1}, …, z_{n}) = (z_{1}^{e_{1}} ⋯ z_{n}^{e_{n}})f(z_{1}, …, z_{n}). First, we must recall a case of the Kunneth theorem for projective resolutions.

Theorem 6. ( Künneth Theorem) Let G be a group and let P: ⋯ → P_{1} →^{d_{1}} P_{0}, Q: ⋯ → Q_{1} →^{δ_{1}} Q_{0} be projective resolutions of ℤ as ℤG modules. Then P ⊗ Q is a projective resolution of ℤ as ℤG modules where [P ⊗ Q]_{n} = ⊕_{i+j=n}P_{i} ⊗ Q_{j}. The boundary maps ∂_{n} are given by ∂_{n}(p_{i} ⊗ q_{j}) = d_{i}(p_{i}) ⊗ q_{j} + (−1)^{j}p_{i} ⊗ δ_{j}(q_{j}) on simple tensors p_{i} ⊗ q_{j}.

Proof: This is Theorem V.2.1 of [4]. □

We aim to give an explicit description of the groups H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) and to show that there is a natural sequence of imbeddings

0 → H^{1}(ℤ, C_{red}^{∗}(ℤ)) → H^{2}(ℤ^{2}, C_{red}^{∗}(ℤ^{2})) → H^{3}(ℤ^{3}, C_{red}^{∗}(ℤ^{3})) → …

In order to do so, we must prove the following lemma.

Lemma 3. Let G = ℤ^{n} be the free abelian group on X_{1}, …, X_{n}. Then there exists a projective resolution of ℤ as ℤG modules with exactly n+1 nonzero modules (including ℤ): 0 → ℤG →^{ρ_{n}} (ℤG)^{n} → ⋯ → ℤ → 0 where the map ρ_{n} is given by ρ_{n}(1) = [(X_{1} − 1), −(X_{2} − 1), …, −(X_{n} − 1)].

Proof: Note that for all n, ℤG = ℤ[X_{1}, …, X_{n}, X_{1}^{−1}, …, X_{n}^{−1}]. For n = 1, we know that there is a projective resolution of ℤ: 0 → ℤG →^{d} ℤG → ℤ → 0, where d is given by d(f(X_{1})) = (X_{1} − 1)f(X_{1}). Suppose that n is at least 2.

Next we must note that for any n, ℤG ≅ ℤℤ ⊗_{ℤ} ℤℤ ⊗_{ℤ} ⋯ ⊗_{ℤ} ℤℤ where there are n terms, and the isomorphism is an isomorphism of ℤG modules. The isomorphism θ: ℤG → ℤℤ ⊗_{ℤ} ⋯ ⊗_{ℤ} ℤℤ is given explicitly by 1 ⟼ 1 ⊗ ⋯ ⊗ 1. The action of G (the free abelian group on X_{1}, …, X_{n}) on ℤℤ ⊗_{ℤ} ⋯ ⊗_{ℤ} ℤℤ is as follows:

X_{1}^{r_{1}} ⋯ X_{n}^{r_{n}} ∗ a_{1} ⊗ ⋯ ⊗ a_{n} = X_{1}^{r_{1}}a_{1} ⊗ ⋯ ⊗ X_{n}^{r_{n}}a_{n}.

Let us look again at the projective resolution of ℤ as a ℤG module for n = 1 given by 0 → ℤG →^{d} ℤG →^{aug} ℤ → 0. In order to properly view this resolution as a resolution, we will call the first ℤG P_{1} and the second P_{0}. Thus, we have a resolution

0 → P_{1} →^{d} P_{0} →^{aug} ℤ → 0, where P_{1} = P_{0} = ℤℤ.

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 15

The Künneth theorem tells us that the tensor product of resolutions of ℤ is again a resolution, so by induction on n, for all n at least 2, there is a resolution of ℤ:

0 → (P_{1} ⊗_{ℤ} ⋯ ⊗_{ℤ} P_{1}) →^{d_{n}} (P_{1} ⊗_{ℤ} P_{1} ⊗_{ℤ} ⋯ ⊗_{ℤ} P_{0}) ⊕ ⋯ ⊕ (P_{0} ⊗_{ℤ} P_{1} ⊗_{ℤ} ⋯ ⊗_{ℤ} P_{1}) →

⋯ → ℤ → 0

where everywhere … appears implies that there are exactly n objects being either ⊕ed or ⊗ed together, except the last … which implies that there are exactly n + 1 (including ℤ) nonzero terms in the sequence. The map d_{n} is given explicitly from the definition of tensor products of resolutions by

d_{n}(a_{1} ⊗ ⋯ ⊗ a_{n}) =

[(X_{1}−1)a_{n}⊗a_{2}⊗⋯⊗a_{n}, −(a_{1}⊗(X_{2}−1)a_{2}⊗⋯⊗a_{n}), …, −(a_{1}⊗⋯⊗a_{n−1}⊗(X_{n}−1)a_{n})].

Note that the map d_{n} is just a generalization of our original map d : P_{1} → P_{0}. We know that

(P_{1} ⊗_{ℤ} ⋯ ⊗_{ℤ} P_{1} ⊗_{ℤ} P_{0}) ⊕ ⋯ ⊕ (P_{0} ⊗_{ℤ} P_{1} ⊗_{ℤ} ⋯ ⊗_{ℤ} P_{1}) ≅ (ℤG)^{n}

via a map ζ where the isomorphism is as ℤG modules (and where G is still the free abelian group on X_{1}, …, X_{n}). The map ζ sends d_{n}(1 ⊗ ⋯ ⊗ 1) = [((X_{1} − 1) ⊗ 1 ⊗ ⋯ ⊗ 1), −(1 ⊗ (X_{2} − 1) ⊗ ⋯ ⊗ 1), − …, −(1 ⊗ ⋯ ⊗ 1 ⊗ (X_{n} − 1))] to [X_{1}−1, −(X_{2}−1), …, −(X_{n}−1)]. Translating back into ℤG lingo, it follows directly from the Künneth Theorem that we have a projective resolution of ℤ as ℤG modules with exactly n + 1 (including ℤ) nonzero modules

0 → ℤG →^{ρ_{n}} (ℤG)^{n} → ⋯ → ℤ → 0.

The map ρ_{n} : ℤG → (ℤG)^{n} is then of course given by a composition of maps:

ρ_{n}(1) = ζ(d_{n}(θ(1))) = ζ(d_{n}(1 ⊗ ⋯ ⊗ 1)) = [X_{1} − 1, −(X_{2} − 1), …, −(X_{n} − 1)]. □

Now we are able to give an explicit description of the groups H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})).

Theorem 7. Let G = ℤ^{n} be the free abelian group on X_{1}, …, X_{n}. Then we have that

H^{n}(G, C_{red}^{∗}(G)) ≅ \frac{C(𝕋^{n})}{(Z_{1} − 1)C(𝕋^{n}) + ⋯ + (Z_{n} − 1)C(𝕋^{n})}.

Proof: Since for all n, C_{red}^{∗}(ℤ^{n}) ≅ C(𝕋^{n}), it suffices to prove the statement for H^{n}(G, C(𝕋^{n})) in the place of H^{n}(G, C_{red}^{∗}(G)). From the previous lemma, we have a projective resolution of ℤ with n + 1 nonzero terms (including ℤ):

0 → ℤG →^{ρ_{n}} (ℤG)^{n} → ⋯ → ℤ → 0.

From this, we get a new sequence B:

0 → Hom_{ℤG}(ℤ, C(𝕋^{n})) → ⋯ → Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n})) →^{ρ_{n}^{∗}} Hom_{ℤG}(ℤG, C(𝕋^{n})) → 0.

Using this sequence, we may compute the cohomology group

H^{n}(G, C_{red}^{∗}(G)) = H^{n}(G, C(𝕋^{n})) = \frac{Hom_{ℤG}(ℤG, C(𝕋^{n}))}{ρ_{n}^{∗}(Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n})))}.

Define θ: Hom_{ℤG}(ℤG, C(𝕋^{n})) → C(𝕋^{n})/[(Z_{1} − 1)C(𝕋^{n}) + ⋯ + (Z_{n} − 1)C(𝕋^{n})] by ϕ ⟼ ϕ(1) + [(Z_{1} − 1)C(𝕋^{n}) + ⋯ + (Z_{n} − 1)C(𝕋^{n})]. This map is certainly onto, because if F ∈ C(𝕋^{n}), we may just define σ ∈ Hom_{ℤG}(ℤG, C(𝕋^{n})) by σ(1) = F. Now we can compute ker(θ).

Suppose that ϕ ∈ Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n})). Then

ϕ(ρ_{n}(1)) = ϕ(X_{1} − 1, −(X_{2} − 1), …, −(X_{n} − 1)) =
(X_{1} − 1) ∗ ϕ(1, 0, …, 0) + ⋯ + (X_{n} − 1) ∗ ϕ(0, …, 0, −1) ∈
∈ [(Z_{1} − 1)C(𝕋^{n}) + ⋯ + (Z_{n} − 1)C(𝕋^{n})].
⟹ ρ_{n}^{∗}(Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n}))) ⊂ ker(θ).

Now suppose that ϕ ∈ ker(θ). So ϕ(1) = (Z_{1}−1)f_{1}+⋯+(Z_{n}−1)f_{n} for some f_{i} ∈ C(𝕋^{n}). Define π ∈ Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n})) by π(1, 0, …, 0) = f_{1}, π(0, 1, 0, …, 0) =

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 17

−f_{2}, …, π(0, …, 0, 1) = −f_{n}. Then π(ρ_{n}(1)) = ϕ(1), and therefore we have that ϕ = π(ρ_{n}) ∈ ρ_{n}^{∗}(Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n}))).

⟹ H^{n}(G, C_{red}^{∗}(G)) = H^{n}(G, C(𝕋^{n})) =

\frac{Hom_{ℤG}(ℤG, C(𝕋^{n}))}{ρ_{n}^{∗}(Hom_{ℤG}((ℤG)^{n}, C(𝕋^{n})))} ≅ \frac{C(𝕋^{n})}{(Z_{1} − 1)C(𝕋^{n}) + ⋯ + (Z_{n} − 1)C(𝕋^{n})}. □

Theorem 8. There is a sequence of group (and ℂ algebra) imbeddings

0 → H^{1}(ℤ, C_{red}^{∗}(ℤ)) →^{d_{1}} H^{2}(ℤ^{2}, C_{red}^{∗}(ℤ^{2})) →^{d_{2}} H^{3}(ℤ^{3}, C_{red}^{∗}(ℤ^{3})) → …

Proof: By the previous theorem, for all n ⩾ 1,

H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = \frac{C(𝕋^{n})}{(z_{1} − 1)C(𝕋^{n}) + ⋯ + (z_{n} − 1)C(𝕋^{n})}.

Fix an n and define

ϕ: \frac{C(𝕋^{n})}{(z_{1} − 1)C(𝕋^{n}) + ⋯ + (z_{n} − 1)C(𝕋^{n})}

→ \frac{C(𝕋^{n+1})}{(z_{1} − 1)C(𝕋^{n+1}) + ⋯ + (z_{n+1} − 1)C(𝕋^{n+1})}

via f + [(z_{1} − 1)C(𝕋^{n}) + ⋯ + (z_{n} − 1)C(𝕋^{n})] ⟼ f + [(z_{1} − 1)C(𝕋^{n+1}) + ⋯ + (z_{n+1} − 1)C(𝕋^{n+1})].

Since (z_{1}−1)C(𝕋^{n})+⋯+(z_{n}−1)C(𝕋^{n}) ⊂ (z_{1}−1)C(𝕋^{n+1})+⋯+(z_{n+1}−1)C(𝕋^{n+1}), ϕ is certainly well defined. To see that ϕ is injective, we just let n = 1 (the other cases are identical). If f ∈ C(𝕋) and f(z_{1}) = (z_{1} − 1)f_{1}(z_{1}, z_{2}) + (z_{2} − 1)f_{2}(z_{1}, z_{2}) with f_{1}, f_{2} ∈ C(𝕋^{2}), then since f doesn’t vary at all with z_{2}, it follows that we can set z_{2} = 1 and then we have that f(z_{1}) = (z_{1} − 1)f(z_{1}, 1) ∈ (z_{1} − 1)C(𝕋). Thus, ker(ϕ) = 0 and so ϕ is injective. □

Theorem 9. For n ⩾ 1,

dim_{ℂ} H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = ∞.

18 ANTHONY J. NARKAWICZ (VIRGINIA TECH)

Proof: For all n, by the last theorem H^{1}(ℤ, C_{red}^{∗}(ℤ)) imbeds in H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})). Since dim_{ℂ} H^{1}(ℤ, C_{red}^{∗}(ℤ)) = ∞, the result follows. □

Finally, using Theorem 4, we are able give a more simple proof of the following result (one proof is given above in Theorem 5).

Theorem 10. H^{1}(ℤ, C_{red}^{∗}(ℤ)) is not Hausdorff and dim_{ℂ} H^{1}(ℤ, C_{red}^{∗}(ℤ)) = ∞.

Proof: We know that for α ∈ ℂℤ, ||α||_{1} ⩾ ||α|_{op} ⩾ ||α||_{2}. Thus, Theorem 4 tells us that H^{1}(ℤ, C_{red}^{∗}(ℤ)) is not Hausdorff. □

5. Our Results (Summarized)

Let G be an infinite, finitely generated group. Let M be a Banach space (with norm || ||_{M}) that is also a left ℂG module such that G acts on M as continuous ℂ-linear transformations. In summary, the following useful results were proved in this paper and do not appear in any papers to date.

(1) Suppose that ℂG ⊂ M ⊂ L^{p}(G) for some p ∈ ℕ. Then the G-module imbedding ℂG → M induces an imbedding of groups H^{1}(G, ℂG) → H^{1}(G, M).

(2) If ℂG ⊂ M ⊂ L^{p}(G) for some p ∈ ℕ and if H^{1}(G, M) = 0, then G has exactly 1 end.

(3) H^{1}(G, M) is not Hausdorff if and only if there exists f_{i} ∈ M with norm 1 (||f_{i}||_{M} = 1) for all i with the property that ||gf_{i} − f_{i}||_{M} ⟶ 0 as i ⟶ ∞ for every g ∈ G.

(4) If ℂG ⊂ M ⊂ L^{p}(G) for some p ⩾ 2, ||α||_{1} ⩾ ||α||_{M} ⩾ ||α||_{p} for every α ∈ ℂG, and if G satisfies the strong Følner condition, then H^{1}(G, M) is not Hausdorff and therefore dim_{ℂ} H^{1}(G, M) = ∞.

We have used these results (with our motivation coming from Effros’s paper ([2])) to show the following for n ⩾ 1.

(1) H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) is not Hausdorff and thus dim_{ℂ} H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = ∞ (with two proofs for n = 1).

THE FIRST COHOMOLOGY GROUP H^{1}(G, M) 19

(2) dim_{ℂ} H^{n}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = ∞.

(3) There is a natural sequence of imbeddings

0 → H^{1}(ℤ, C_{red}^{∗}(ℤ)) → H^{2}(ℤ^{2}, C_{red}^{∗}(ℤ^{2})) → H^{3}(ℤ^{3}, C_{red}^{∗}(ℤ^{3})) → …

(4)

H^{1}(ℤ^{n}, C_{red}^{∗}(ℤ^{n})) = \frac{C(𝕋^{n})}{(Z_{1} − 1)C(𝕋^{n}) + ⋯ + (Z_{n} − 1)C(𝕋^{n})}.

6. Conjectures and Future Work

Conjecture 1. Let G be an infinite, finitely generated group. Let M be a Banach space (with norm || ||_{M}) that is also a left ℂG module such that G acts on M as continuous ℂ-linear transformations. Then G is amenable if and only if H^{1}(G, M) is not Hausdorff.

We should note that the proof given in [3] of this with L^{2}(G) is specific to that case.

References

[1] Alain Guichardet: Cohomologie des groupes topologiques et des algebres de Lie, Cedic-Fernand Nathan,Paris,1980.

[2] Edward G. Effros: Why the Circle Is Connected: An Introduction to Quantized Topology, The Math. Intelligencer 15 (1982), 197–211.

[3] Mohammed E.B. Bekka and Alain Valette: Group Cohomology, Harmonic Functions and the First L^{2}-Betti Number, Potential Anal. 6 (1997), 313-326.

[4] P.J. Hilton and U. Stammbach: A Course in Homological Algebra, Springer-Verlag, New York, 1997. Graduate Texts in Mathematics, 4.

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