EXCISION IN BANACH SIMPLICIAL AND CYCLIC COHOMOLOGY

ZINAIDA A. LYKOVA

Abstract. We prove that, for every extension of Banach algebras 0 → B → A → D → 0 such that B has a left or right bounded approximate identity, the existence of an associated long exact sequence of Banach simplicial or cyclic cohomology groups is equivalent to the existence of one for homology groups. It follows from the continuous version of a result of Wodzicki that associated long exact sequences exist. In particular, they exist for every extension of C^{∗}-algebras.

1. Introduction

Significant results of A. Connes on non-commutative differential geometry [Co1] have led much research interest to the computation of cyclic (co)homology groups in recent years; see, for example, [Lo] and [Co2] for many references and [ChS], [Gr], [He2], [Wo1] and [Wo2] for the continuous theory of these groups. A promising approach to the calculation of cyclic cohomology groups is to break it down by making use of extensions of Banach algebras; this is a standard device in the study of various properties of Banach algebras. Here we establish (Theorem 3.5) that, for every weakly admissible extension of Banach algebras 0 → B → A → D → 0, the existence of an associated long exact sequence of Banach simplicial cohomology groups is equivalent to the existence of one for homology groups. Theorem 3.8 shows that the same result is true for cyclic (co)homology groups when B has a left or right bounded approximate identity. Recall that the existence of long exact sequences of simplicial and cyclic homology groups associated with an extension of algebras (as distinct from cohomology and, mainly, in an algebraic context) has been studied by M. Wodzicki [Wo2], [Wo3]; see also [Lo]. Wodzicki remarked that his result could easily be extended to the continuous case under some hypotheses on the extension (see Remark (3) and Corollary 4 [Wo2] and Remark 8.5(2) [Wo3]).

From Theorems 3.5, 3.8 and the continuous version of Wodzicki’s result (Theorem 4.1) for homology groups we deduce the existence of long exact sequences of simplicial and cyclic cohomology groups for every extension when B has a left or right bounded approximate identity. In particular, this is true for every extension of C^{∗}-algebras. This gives an effective tool for computing Banach simplicial and cyclic cohomology groups (see Proposition 4.4). We apply it to some natural classes of Banach algebras.

Key words and phrases. Cohomology of Banach algebras.

I am indebted to the Mathematics Department of the University of California and the Mathematical Sciences Research Institute at Berkeley for hospitality while this work was carried out.

Research at MSRI is supported in part by NSF grant DMS-9022140.

Finally, I am grateful to M.Wodzicki for suggestions about the continuous homology of Banach algebras.

2. Definitions and notation

We recall some notation and terminology used in the homological theory of Banach algebras.

Throughout the paper id denotes the identity operator. We denote the projective tensor product of Banach spaces by ˆ⊗ (see, for example, [Pa]). Note that by Z^{ˆ⊗0} ˆ⊗Y we mean Y and by Z^{ˆ⊗1} we mean Z.

Let A be a Banach algebra, not necessarily unital. We will define the Banach version of the cyclic homology HC_{n}(A) (see, for example, [Lo], [Wo2] or [He2]). We denote by C_{n}(A), n = 0, 1, ..., the (n + 1) fold projective tensor power A^{ˆ⊗(n+1)} = A ˆ⊗ … ˆ⊗A of A; we shall call the elements of this Banach space n-dimensional chains. We let t\underline{n} : C_{n}(A) → C_{n}(A), n = 0, 1, ..., denote the operator given by t\underline{n}(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n}) = (−1)^{n}(a_{n} ⊗ a_{0} ⊗ ⋯ ⊗ a_{n−1}), and we set t\underline{0} = id. We let CC_{n}(A) denote the quotient space of C_{n}(A) modulo the closure of the linear span of elements of the form x − t\underline{n}x where n = 0, 1, …. Note that, by Proposition 4 [He2], Im (id_{C_{n}(A)} − t\underline{n}) is closed in C_{n}(A) and so CC_{n}(A) = C_{n}(A)/Im (id_{C_{n}(A)} − t\underline{n}). We also set CC_{0}(A) = C_{0}(A) = A.

From the chains we form the standard homology complex

0 ← C_{0}(A) ←^{d_{0}} ⋯ ← C_{n}(A) ←^{d_{n}} C_{n+1}(A) ← …, (C_{∼}(A))

where the differential d_{n} is given by the formula

d_{n}(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n+1}) =

∑_{i=0}^{n}(−1)^{i}(a_{0} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1}) + (−1)^{n+1}(a_{n+1}a_{0} ⊗ ⋯ ⊗ a_{n}).

It is not difficult to verify that these d_{n} induce operators dc_{n} : CC_{n+1}(A) → CC_{n}(A) in the respective quotient spaces. Thus we obtain a quotient complex CC_{∼}(A) of the complex C_{∼}(A). The n-dimensional homology of C_{∼}(A), denoted by H_{n}(A), is called the n-dimensional Banach simplicial homology group of the Banach algebra A. The n-dimensional homology of CC_{∼}(A), denoted by HC_{n}(A), is called the n-dimensional Banach cyclic homology group of A.

We also consider the complex

0 ←^{dr_{−1}} C_{0}(A) ←^{dr_{0}} ⋯ ← C_{n}(A) ←^{dr_{n}} C_{n+1}(A) ← …, (CR_{∼}(A))

where the differential dr_{n} is given by the formula

dr_{n}(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n+1}) = ∑_{i=0}^{n}(−1)^{i}(a_{0} ⊗ ⋯ ⊗ a_{i}a_{i+1} ⊗ ⋯ ⊗ a_{n+1})

and we set dr_{−1}(a_{0}) = 0. The n-dimensional homology of CR_{∼}(A), denoted by HR_{n}(A), is called the n-dimensional Banach Bar homology group of A.

Note that, by definition, dc_{0} = d_{0}, so that HC_{0}(A) = H_{0}(A) = A/Im d_{0}.

Obviously H_{n}(A) is just another way of writing the Hochschild homology group H_{n}(A, A) (see [He1; II.5.5]).

For a Banach A-bimodule X, we will denote the n-dimensional Banach cohomology group of A with coefficients in X by H^{n}(A, X) (see, for example, [Jo1] or [He1]). Recall that a Banach A-bimodule M = (M_{∗})^{∗}, where M_{∗} is a Banach A-bimodule, is called dual. Here E^{∗} is the dual Banach space of a Banach space E. A Banach algebra A such that H^{1}(A, M) = {0} for all dual A-bimodules M is called amenable.

The nth cohomology of the dual complex C^{∼}(A) =^{def} C_{∼}(A)^{∗}, denoted by H^{n}(A), is called the n-dimensional Banach simplicial cohomology group of the Banach algebra A. The nth cohomology of the dual complex CC^{∼}(A) =^{def} CC_{∼}(A)^{∗}, denoted by HC^{n}(A), is called the n-dimensional Banach cyclic cohomology group of A (see [Co1] or [He2]). The nth cohomology of the dual complex CR^{∼}(A) =^{def} CR_{∼}(A)^{∗}, denoted by HR^{n}(A), is called the n-dimensional Banach Bar cohomology group of the Banach algebra A.

Note that, by definition, HC^{0}(A) = H^{0}(A) coincides with the space A^{tr} = {f ∈ A^{∗} : f(ab) = f(ba) for all a, b ∈ A} of continuous traces on A.

The canonical identification of (n + 1)−linear functionals on A and n−linear operators from A to A^{∗} shows that H^{n}(A) is just another way of writing H^{n}(A, A^{∗}).

The vanishing of HR^{n}(A) for a Banach algebra A implies the existence of the Connes-Tsygan exact sequence for A. One can find the conditions for this in Theorems 15 and 16 [He2].

The short exact sequence of Banach spaces and continuous linear operators

0 → Y →^{i} Z →^{j} W → 0

is called admissible if there exists a continuous operator α : W → Z such that j ∘ α = id_{W}. Recall that admissibility is equivalent to the existence of a continuous operator β : Z → Y such that β ∘ i = id_{Y}. This is also equivalent to the existence of continuous operators β : Z → Y and α : W → Z such that β ∘ i = id_{Y}, j ∘ α = id_{W} and i ∘ β + α ∘ j = id_{Z}.

The short exact sequence of Banach spaces and continuous linear operators 0 → Y →^{i} Z →^{j} W → 0 is called weakly admissible if the dual short sequence 0 → W^{∗} →^{j^{∗}} Z^{∗} →^{i^{∗}} Y^{∗} → 0 is admissible.

Definitions of a (co)chain complex, a morphism of complexes, the homology groups can be found in any text book on homological algebra, for instance MacLane [Ma], Helemskii [He1] (continuous case), Loday [Lo].

A chain complex K_{∼} in the category of [Banach] linear spaces is a sequence of [Banach] linear spaces and [continuous] linear operators

⋯ ← K_{n} ←^{d_{n}} K_{n+1} ←^{d_{n+1}} K_{n+2} ← …

such that d_{n} ∘ d_{n+1} = 0 for every n. The cycles are the elements of Z_{n} = Ker (d_{n−1} : K_{n} → K_{n−1}). The boundaries are the elements of B_{n} = Im (d_{n} : K_{n+1} → K_{n}). The relation d_{n−1} ∘ d_{n} = 0 implies B_{n} ⊂ Z_{n}. The homology groups are defined by H_{n}(K_{∼}) = Z_{n}/B_{n}.

A [continuous] morphism of chain complexes ψ_{∼} : K_{∼} → P_{∼} in the category of [Banach] linear spaces is a collection of [continuous] linear operators ψ_{n} : K_{n} → P_{n} such that the following diagram is commutative for any n

K_{n} →^{d_{n−1}} K_{n−1}
↓ ψ_{n}  ↓ ψ_{n−1}
P_{n} →^{d′_{n−1}} P_{n−1}.

Such a morphism obviously induces a map H_{n}(ψ_{∼}) : H_{n}(K_{∼}) → H_{n}(P_{∼}).

A [continuous] morphism of chain complexes ψ_{∼} : K_{∼} → P_{∼} of [Banach] linear spaces is a [topological] quasi-isomorphism if H_{n}(ψ_{∼}) : H_{n}(K_{∼}) → H_{n}(P_{∼}) is a [topological] isomorphism for every n.

3. Connections between long exact sequences of homology and cohomology groups

To start with we need the following elementary result from the homological algebra. 3.1 Lemma. Let

⋯ → L_{n−2} →^{ζ_{n−1}} L_{n−1} →^{ζ_{n}} L_{n} →^{ζ_{n+1}} L_{n+1} →^{ζ_{n+2}} L_{n+2} …,

be an exact sequence of linear spaces and linear operators. Suppose that ζ_{n−1} and ζ_{n+2} are isomorphisms. Then L_{n} = {0}.

Proof. By the assumption that ζ_{n−1} is an isomorphism, so Im ζ_{n−1} = L_{n−1} = Ker ζ_{n}. Thus Im ζ_{n} = {0} and, by virtue of the exactness of the sequence, Ker ζ_{n+1} = Im ζ_{n} = {0}. By the assumption that ζ_{n+2} is an isomorphism, Ker ζ_{n+2} = {0} = Im ζ_{n+1}. Therefore, since Ker ζ_{n+1} = {0} and Im ζ_{n+1} = {0}, the result follows.

3.2 Proposition. Let K_{∼} and P_{∼} be chain complexes of Banach spaces and continuous linear operators. Suppose there is a continuous morphism of chain complexes ψ_{∼} : K_{∼} → P_{∼} such that, for each n, ψ_{n} is injective and Imψ_{n} is closed. Then

(I) for fixed n, if H^{k}(ψ_{∼}^{∗}) is an isomorphism whenever n + 2 ≥ k ≥ n − 1, then H_{n}(ψ_{∼}) is a topological isomorphism.

(II) for fixed n, if H_{k}(ψ_{∼}) is an isomorphism whenever n + 1 ≥ k ≥ n − 2, then H^{n}(ψ_{∼}^{∗}) is a topological isomorphism.

(III) the following conditions are equivalent:

(i) ψ_{∼} is a topological quasi-isomorphism of the chain complexes K_{∼} and P_{∼}; (ii) ψ_{∼}^{∗} is a topological quasi-isomorphism of the cochain, dual complexes K_{∼}^{∗} and

P_{∼}^{∗}.

Proof. By the assumption, there is a short exact sequence of complexes

0 → K_{∼} →^{ψ_{∼}} P_{∼} →^{ϕ_{∼}} L_{∼} → 0

in the category of Banach spaces and continuous linear operators, where L_{∼} is the chain complex P_{∼}/Imψ_{∼} and ϕ_{n} : P_{n} → P_{n}/Imψ_{n} is the quotient mapping. Hence, by [He1, Theorem 0.5.7], there exists a long exact sequence

⋯ → H_{n+1}(L_{∼}) →^{ζ_{n+1}} H_{n}(K_{∼}) →^{H_{n}(ψ_{∼})} H_{n}(P_{∼}) →^{H_{n}(ϕ_{∼})} H_{n}(L_{∼}) →^{ζ_{n}} H_{n−1}(K_{∼}) → … (3.1)

of homology groups, where ζ_{n} (the connecting morphism), H_{n}(ϕ_{∼}) and H_{n}(ψ_{∼}) are continuous linear operators.

We can also consider a short sequence of dual cochain complexes

0 → L_{∼}^{∗} →^{ϕ_{∼}^{∗}} P_{∼}^{∗} →^{ψ_{∼}^{∗}} K_{∼}^{∗} → 0

in the category of Banach spaces and continuous linear operators. This sequence is exact by [Jo1; Section 1] (see also, [He1; 0.5.2]). Hence, by [He1, Theorem 0.5.7], there exists a long exact sequence

⋯ → H^{n−1}(K_{∼}^{∗}) →^{ξ_{n}} H^{n}(L_{∼}^{∗}) →^{H^{n}(ϕ_{∼}^{∗})} H^{n}(P_{∼}^{∗}) →^{H^{n}(ψ_{∼}^{∗})} H^{n}(K_{∼}^{∗}) →^{ξ_{n+1}} H^{n+1}(L_{∼}^{∗}) … (3.2)

of cohomology groups, where ξ_{n} (the connecting morphism), H^{n}(ϕ_{∼}^{∗}) and H^{n}(ψ_{∼}^{∗}) are continuous linear operators.

(I) Since H^{k}(ψ_{∼}^{∗}) is an isomorphism whenever n + 2 ≥ k ≥ n − 1, by Lemma 3.1, H^{k}(L_{∼}^{∗}) = {0} whenever n + 2 ≥ k ≥ n. Thus, by [Jo1; Section 1], H_{k}(L_{∼}) = {0} for k = n and k = n + 1, and so H_{n}(ψ_{∼}) is an isomorphism. Hence, by Lemma 7.1.32 [BS], H_{n}(ψ_{∼}) is a topological isomorphism.

(II) Since H_{k}(ψ_{∼}) is an isomorphism whenever n + 1 ≥ k ≥ n − 2, by Lemma 3.1, H_{k}(L_{∼}) = {0} whenever n + 1 ≥ k ≥ n − 1. Thus, by [Jo1; Section 1], H^{k}(L_{∼}^{∗}) = {0} for k = n and k = n + 1, and so H^{n}(ψ_{∼}^{∗}) is an isomorphism. Hence, by Lemma 7.1.32 [BS], H^{n}(ψ_{∼}^{∗}) is a topological isomorphism.

(III) Recall that a complex L_{∼} is exact if and only if its dual complex L_{∼}^{∗} is exact [Jo1; Section 1] (see also [He1; 0.5.2]). Thus the homology groups H_{n}(L_{∼}) vanish if and only if the cohomology groups H^{n}(L_{∼}^{∗}) vanish.

Since the sequences (3.1) and (3.2) are exact, the result follows from Lemma 3.1 and Theorem 0.5.10 [He1].

The following lemma is widely known.

3.3 Lemma. Let

0 → Y →^{i} Z →^{j} W → 0  (3.3)

be a weakly admissible sequence of Banach spaces and continuous operators. Then, for every Banach space X, the sequence

0 → X ˆ⊗Y →^{id_{X}⊗i} X ˆ⊗Z →^{id_{X}⊗j} X ˆ⊗W → 0

is weakly admissible.

Proof. Since the sequence (3.3) is weakly admissible, there are continuous linear operators α : Y^{∗} → Z^{∗} and β : Z^{∗} → W^{∗} such that i^{∗} ∘ α = id_{Y^{∗}}, β ∘ j^{∗} = id_{W^{∗}} and α ∘ i^{∗} + j^{∗} ∘ β = id_{Z^{∗}}.

Further we will use the well-known isomorphism (see, for example, [He1; Theorem 2.2.17])

B(X, Y^{∗}) → (X ˆ⊗Y)^{∗} : ϕ ↦ Φ_{ϕ}

where Φ_{ϕ}(x ⊗ y) = [ϕ(x)](y); x ∈ X, y ∈ Y and

(X ˆ⊗Y)^{∗} → B(X, Y^{∗}) : f ↦ ϕ_{f}

where [ϕ_{f}(x)](y) = f(x ⊗ y); x ∈ X, y ∈ Y.

We can see that, for f ∈ (X ˆ⊗Y)^{∗}, we have ϕ_{f} ∈ B(X, Y^{∗}), α ∘ ϕ_{f} ∈ B(X, Z^{∗}) and Φ_{α∘ϕ_{f}} ∈ (X ˆ⊗Z)^{∗}. We define γ(f) = Φ_{α∘ϕ_{f}} for f ∈ (X ˆ⊗Y)^{∗} and η(g) = Φ_{β∘ϕ_{g}} for g ∈ (X ˆ⊗Z)^{∗}.

We can check that (id_{X} ⊗ i)^{∗} ∘ γ = id_{(X ˆ⊗Y)^{∗}}, η ∘ (id_{X} ⊗ j)^{∗} = id_{(X ˆ⊗W)^{∗}} and γ ∘ (id_{X} ⊗ i)^{∗} + (id_{X} ⊗ j)^{∗} ∘ η = id_{(X ˆ⊗Z)^{∗}}. The result follows.

3.4 Lemma. Let

0 → Y →^{i} Z →^{j} W → 0

be a weakly admissible sequence of Banach spaces and continuous operators. Then, for every n, the following conditions are satisfied

(i) the operator

i^{ˆ⊗n} = i ˆ⊗ … ˆ⊗i : Y ˆ⊗ … ˆ⊗Y → Z ˆ⊗ … ˆ⊗Z

is injective and Im(i^{ˆ⊗n}) is closed;

(ii) Ker j^{ˆ⊗n} = Z^{ˆ⊗(n−1)} ˆ⊗i(Y) + Z^{ˆ⊗(n−2)} ˆ⊗i(Y) ˆ⊗Z + ⋯ + i(Y) ˆ⊗Z^{ˆ⊗(n−1)} and so it is the closure of linear span {z_{1} ⊗ z_{2} ⊗ ⋯ ⊗ z_{n}: at least one z_{i} belongs to i(Y)}.

(iii) the complex

0 → Y^{ˆ⊗n} →^{i^{ˆ⊗n}} Z^{ˆ⊗n} →^{j^{ˆ⊗n}} (W)^{ˆ⊗n} → 0

is exact at Y^{ˆ⊗n} and (W)^{ˆ⊗n}; Proof. (i) We can see that

i^{ˆ⊗n} = κ_{n} ∘ (id_{Z^{ˆ⊗(n−1)}} ⊗ i) ∘ κ_{n−1} ∘ ⋯ ∘ (id_{Z ˆ⊗Y^{ˆ⊗(n−2)}} ⊗ i) ∘ κ_{1} ∘ (id_{Y^{ˆ⊗(n−1)}} ⊗ i),

where κ_{i} : Z^{ˆ⊗(i−1)} ˆ⊗Y^{ˆ⊗(n−i)} ˆ⊗Z ≅ Z^{ˆ⊗i} ˆ⊗Y^{ˆ⊗(n−i)}, i = 1, …, n are given by κ_{i}(x_{1} ⊗ x_{2} ⊗ ⋯ ⊗ x_{n}) = x_{n} ⊗ x_{1} ⊗ ⋯ ⊗ x_{n−1}.

Thus

(i^{ˆ⊗n})^{∗} = (id_{Y^{ˆ⊗(n−1)}} ⊗i)^{∗} ∘(κ_{1})^{∗} ∘(id_{Z⊗Y^{ˆ⊗(n−2)}} ⊗i)^{∗} ∘⋯∘(κ_{n−1})^{∗} ∘(id_{Z^{ˆ⊗(n−1)}} ⊗i)^{∗} ∘(κ_{n})^{∗}.

By Lemma 3.3, (i^{ˆ⊗n})^{∗} is surjective as the composition of surjective operators. Therefore, by [Ed; 8.6.15], i^{ˆ⊗n} is injective and Im(i^{ˆ⊗n}) is closed in Z^{ˆ⊗n}.

(ii) We can see that

j^{ˆ⊗n} = ψ_{1} ∘ ψ_{2} ∘ ⋯ ∘ ψ_{n}

where ψ_{1} = j ⊗ id_{W^{ˆ⊗(n−1)}}, ψ_{i} = id_{Z^{ˆ⊗(i−1)}} ⊗ j ⊗ id_{W^{ˆ⊗(n−i)}} for 1 < i < n and ψ_{n} = id_{Z^{ˆ⊗(n−1)}} ⊗ j. By Lemma 3.3, we have Ker ψ_{i} = Im id_{Z^{ˆ⊗(i−1)}} ⊗ i ⊗ id_{W^{ˆ⊗(n−i)}} = Z^{ˆ⊗(i−1)} ˆ⊗i(Y) ˆ⊗W^{ˆ⊗(n−i)} and, for F_{i} = Z^{ˆ⊗(i−1)} ˆ⊗i(Y) ˆ⊗Z^{ˆ⊗(n−i)}, we obtain

(ψ_{i+1} ∘ ⋯ ∘ ψ_{n−1} ∘ ψ_{n})(F_{i}) = Ker ψ_{i}.

Thus we have

Ker ψ_{n} = Z^{ˆ⊗(n−1)} ˆ⊗i(Y);
Ker (ψ_{n−1} ∘ ψ_{n}) = ψ_{n}^{−1}(Ker ψ_{n−1}) =
Ker ψ_{n} + F_{n−1} = Z^{ˆ⊗(n−1)} ˆ⊗i(Y) + Z^{ˆ⊗(n−2)} ˆ⊗i(Y) ˆ⊗Z;
Ker (ψ_{n−2} ∘ ψ_{n−1} ∘ ψ_{n}) = (ψ_{n−1} ∘ ψ_{n})^{−1}(Ker ψ_{n−2}) =
Ker (ψ_{n−1} ∘ ψ_{n}) + F_{n−2} =
Z^{ˆ⊗(n−1)} ˆ⊗i(Y) + Z^{ˆ⊗(n−2)} ˆ⊗i(Y) ˆ⊗Z + Z^{ˆ⊗(n−3)} ˆ⊗i(Y) ˆ⊗Z^{ˆ⊗2};
. . . . . .
Ker (ψ_{1} ∘ … ψ_{n−1} ∘ ψ_{n}) = (ψ_{2} ∘ … ψ_{n−1} ∘ ψ_{n})^{−1}(Ker (ψ_{1}) =
Ker (ψ_{2} ∘ … ψ_{n−1} ∘ ψ_{n}) + F_{1} =
Z^{ˆ⊗(n−1)} ˆ⊗i(Y) + Z^{ˆ⊗(n−2)} ˆ⊗i(Y) ˆ⊗Z + ⋯ + i(Y) ˆ⊗Z^{ˆ⊗(n−1)}.

(iii) It follows from (i) and the fact that the projective tensor product of surjective operators is also surjective.

3.5 Theorem. Let

0 → B →^{i} A →^{j} D → 0  (3.4)

be a weakly admissible extension of Banach algebras. Then

(I) there exists an associated long exact sequence of Banach simplicial homology groups

⋯ → H_{n}(B) → H_{n}(A) → H_{n}(D) → H_{n−1}(B) → ⋯ → H_{0}(D) → 0 (3.5)

if and only if there exists an associated long exact sequence of Banach simplicial cohomology groups

0 → H^{0}(D) → ⋯ → H^{n−1}(B) → H^{n}(D) → H^{n}(A) → H^{n}(B) → …; (3.6)

(II) there exists an associated long exact sequence of Banach Bar homology groups

⋯ → HR_{n}(B) → HR_{n}(A) → HR_{n}(D) → HR_{n−1}(B) → ⋯ → HR_{0}(D) → 0 (3.7)

if and only if there exists an associated long exact sequence of Banach Bar cohomology groups

0 → HR^{0}(D) → ⋯ → HR^{n−1}(B) → HR^{n}(D) → HR^{n}(A) → HR^{n}(B) → …. (3.8)

Proof. (I) By Lemma 3.4 (iii), for n = 0, 1, …, the extension (3.4) induces a complex of Banach spaces and continuous operators

0 → C_{n}(B) →^{i^{ˆ⊗(n+1)}} C_{n}(A) →^{j^{ˆ⊗(n+1)}} C_{n}(D) → 0

which is exact at C_{n}(D) and C_{n}(B). It is routine to check that the diagram

0  0  0
↑  ↑  ↑
0 → C_{0}(B) →^{i} C_{0}(A) →^{j} C_{0}(D) → 0
↑ d_{0}  ↑ d_{0}  ↑ d_{0}
0 → C_{1}(B) →^{i ˆ⊗i} C_{1}(A) →^{j ˆ⊗j} C_{1}(D) → 0
↑ d_{1}  ↑ d_{1}  ↑ d_{1}
. . . . . .  …  …  …  …  …  . . . . . .
↑  ↑  ↑
0 → C_{n}(B) →^{i ˆ⊗... ˆ⊗i} C_{n}(A) →^{j ˆ⊗... ˆ⊗j} C_{n}(D) → 0
↑ d_{n}  ↑ d_{n}  ↑ d_{n}
0 → C_{n+1}(B) →^{i ˆ⊗... ˆ⊗i} C_{n+1}(A) →^{j ˆ⊗... ˆ⊗j} C_{n+1}(D) → 0
↑  ↑  ↑
. . . . . .  …  …  …  …  …  . . . . . .

is commutative. The same is true for cochains. Thus the extension (3.4) induces a short exact sequence of chain complexes

0 → C_{∼}(A, D) → C_{∼}(A) → C_{∼}(D) → 0

and a short exact sequence of cochain complexes

0 → C^{∼}(D) → C^{∼}(A) → C^{∼}(A; D) → 0

in the category of Banach spaces and continuous operators, where C_{∼}(A, D) is the subcomplex Ker(j ˆ⊗ … ˆ⊗j) of C_{∼}(A); C^{∼}(A; D) is the subcomplex [Ker(j ˆ⊗ … ˆ⊗j)]^{∗} of C^{∼}(A). Hence, by [He1, Chapter 0, Section 5.5], there exist a long exact sequence of homology groups

⋯ → H_{n+1}(D) → H_{n}(C_{∼}(A; D)) → H_{n}(A) → H_{n}(D) → …

and a long exact sequence of cohomology groups

⋯ → H^{n−1}(C^{∼}(A; D)) → H^{n}(D) → H^{n}(A) → H^{n}(C^{∼}(A; D)) → ….

By Lemma 3.4 (i),

i^{ˆ⊗(n+1)} : B^{ˆ⊗(n+1)} → Ker j^{ˆ⊗(n+1)}

is injective and Im(i^{ˆ⊗(n+1)}) is closed. Therefore, by Proposition 3.2, for the two complexes C_{∼}(B) and C_{∼}(A; D) and the continuous morphism {i^{ˆ⊗(n+1)}} of complexes, we have H_{n}(C_{∼}(A; D)) ≅ H_{n}(B) for n = 0, 1, … if and only if H^{n}(C^{∼}(A; D)) ≅ H^{n}(B) for n = 0, 1, . . .. The result follows from the five lemma [Ma, Lemma 1.3.3].

(II) follows by the same arguments as in the case (I).

The following lemma is widely known.

3.6 Lemma. Let

0 → B →^{i} A →^{j} D → 0  (3.9)

be an extension of Banach algebras. Suppose that B has a left or right bounded approximate identity (e_{α})_{α∈Λ}. Then (3.9) is a weakly admissible.

Proof. Consider the Fréchet filter F on Λ, with base {Q_{λ} : λ ∈ Λ}, where Q_{λ} = {α ∈ Λ : α ≥ λ}. Thus

F = {E ⊂ Λ : there is a λ ∈ Λ such that Q_{λ} ⊂ E}.

Let U be an ultrafilter on Λ which refines F. One can find information on filters in [Bo].

Suppose that (e_{α})_{α∈Λ} is a right bounded approximate identity. For f ∈ B^{∗} we define g_{f} ∈ A^{∗} by

g_{f}(a) = lim_{α→U} f(i^{−1}(ai(e_{α}))) for all a ∈ A.

It is easy to check that g_{f} is a bounded linear functional, the operator

L : B^{∗} → A^{∗} : f ↦ g_{f}

is a bounded linear operator and i^{∗} ∘ L = id_{B^{∗}}.

Let

0 → B →^{i} A →^{j} D → 0

be an extension of Banach algebras. Note that (id_{A^{ˆ⊗(n+1)}} − t\underline{n}) ∘ i^{ˆ⊗(n+1)} = i^{ˆ⊗(n+1)} ∘ (id_{B^{ˆ⊗(n+1)}} − t\underline{n}) and (id_{D^{ˆ⊗(n+1)}} − t\underline{n}) ∘ j^{ˆ⊗(n+1)} = j^{ˆ⊗(n+1)} ∘ (id_{A^{ˆ⊗(n+1)}} − t\underline{n}). Thus there is a complex

0 → CC_{n}(B) →^{{ \atop i^{ˆ⊗(n+1)}}} CC_{n}(A) →^{{ \atop j^{ˆ⊗(n+1)}}} CC_{n}(D) → 0,

where { \atop i^{ˆ⊗(n+1)}} and { \atop j^{ˆ⊗(n+1)}} are induced by i^{ˆ⊗(n+1)} and j^{ˆ⊗(n+1)} respectively.

3.7 Lemma. Let

0 → B →^{i} A →^{j} D → 0

be an extension of Banach algebras. Suppose B has a left or right bounded approximate identity (e_{α})_{α∈Λ}. Then, for every n, the complex

0 → CC_{n}(B) →^{{ \atop i^{ˆ⊗(n+1)}}} CC_{n}(A) →^{{ \atop j^{ˆ⊗(n+1)}}} CC_{n}(D) → 0,

is exact at CC_{n}(B) and CC_{n}(D); { \atop i^{ˆ⊗(n+1)}} is injective and Im({ \atop i^{ˆ⊗(n+1)}}) is closed; and

Ker { \atop j^{ˆ⊗(n+1)}} = \{u+Im (id_{A^{ˆ⊗(n+1)}} −t\underline{n}); where u ∈ Ker j^{ˆ⊗(n+1)}\} and so it is the closure of the linear span of \{a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} + Im (id_{A^{ˆ⊗(n+1)}} − t\underline{n}) : at least one a_{i} belongs to i(B)}.

Proof. Since j^{ˆ⊗(n+1)} is surjective and (id_{D^{ˆ⊗(n+1)}} −t\underline{n})∘j^{ˆ⊗(n+1)} = j^{ˆ⊗(n+1)}∘(id_{A^{ˆ⊗(n+1)}} −

t\underline{n}), the mapping { \atop j^{ˆ⊗(n+1)}} : CC_{n}(A) → CC_{n}(D) is surjective too.

Suppose that (e_{α})_{α∈Λ} is a right bounded approximate identity. Let us check that

({ \atop i^{ˆ⊗(n+1)}})^{∗} is surjective. For

f ∈ CC_{n}(B)^{∗} = \{f ∈ (B^{ˆ⊗(n+1)})^{∗} : f|_{Im (id_{B^{ˆ⊗(n+1)}}−t\underline{n})} = 0\},

we define g_{f} ∈ (A^{ˆ⊗(n+1)})^{∗} by

g_{f}(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n}) = lim_{α→U} f(i^{−1}(a_{0}i(e_{α})) ⊗ ⋯ ⊗ i^{−1}(a_{n}i(e_{α}))),

where a_{i} ∈ A; i = 0, …, n and the ultrafilter U is defined as in Lemma 3.6. It is easy to check that g_{f} is a bounded linear functional, g_{f}|_{Im (id_{A^{ˆ⊗(n+1)}}−t\underline{n})} = 0 and

f = (i^{ˆ⊗(n+1)})^{∗}g_{f}. Thus, by [Ed; 8.6.15], { \atop i^{ˆ⊗(n+1)}} : CC_{n}(B) → CC_{n}(A) is injective and its image is closed.

Now let us show that there exists a continuous operator α_{n} : C_{n}(A)^{∗} → C_{n}(D)^{∗} such that α_{n} ∘ (j^{ˆ⊗(n+1)})^{∗} = id_{C_{n}(D)^{∗}} and α_{n}(CC_{n}(A)^{∗}) ⊂ CC_{n}(D)^{∗}. For g ∈ C_{n}(A)^{∗}, we define f_{g} ∈ C_{n}(D)^{∗}, by

f_{g}(d_{0} ⊗ d_{1} ⊗ ⋯ ⊗ d_{n}) =

g(a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n}) − lim_{α→U} ∑_{k=0}^{n} g(a_{0} ⊗ ⋯ ⊗ a_{k−1} ⊗ a_{k}i(e_{α}) ⊗ a_{k+1} ⊗ ⋯ ⊗ a_{n})+

lim_{α→U} ∑_{0≤k<t≤n} g(a_{0} ⊗ ⋯ ⊗ a_{k−1} ⊗ a_{k}i(e_{α}) ⊗ a_{k+1} ⊗ ⋯ ⊗ a_{t}i(e_{α}) ⊗ a_{t+1} ⊗ ⋯ ⊗ a_{n}) − ⋯ +

(−1)^{n+1} lim_{α→U} g(a_{0}i(e_{α}) ⊗ a_{1}i(e_{α}) ⊗ ⋯ ⊗ a_{n}i(e_{α})),

where d_{i} = j(a_{i}) ∈ D; i = 0, …, n. It is easy to check that the operator

α_{n} : C_{n}(A)^{∗} → C_{n}(D)^{∗} : g ↦ f_{g}

is a bounded linear operator, α_{n}∘(j^{ˆ⊗(n+1)})^{∗} = id_{C_{n}(D)^{∗}} and α_{n}(CC_{n}(A)^{∗}) ⊂ CC_{n}(D)^{∗}. Hence we can see that the dual map of

σ_{n} : Ker j^{ˆ⊗(n+1)} → Ker { \atop j^{ˆ⊗(n+1)}} : u ↦ u + Im (id_{A^{ˆ⊗(n+1)}} − t\underline{n})

is injective and Im σ_{n} is closed. Thus, by [Ed; 8.6.15], σ_{n} is surjective, that is,

(Ker { \atop j^{ˆ⊗(n+1)}})^{∗} = Im σ_{n}. It follows from Lemma 3.4(ii) that the Ker { \atop j^{ˆ⊗(n+1)}} is the closure of the linear span of \{a_{0} ⊗ a_{1} ⊗ ⋯ ⊗ a_{n} + Im (id_{A^{ˆ⊗(n+1)}} − t\underline{n}) : at least one a_{i} belongs to i(B)}.

3.8 Theorem. Let

0 → B →^{i} A →^{j} D → 0

be an extension of Banach algebras. Suppose B has a left or right bounded approximate identity. Then there exists an associated long exact sequence of Banach cyclic homology groups

⋯ → HC_{n}(B) → HC_{n}(A) → HC_{n}(D) → HC_{n−1}(B) → ⋯ → HC_{0}(D) → 0 (3.10)

if and only if there exists an associated long exact sequence of Banach cyclic cohomology groups

0 → HC^{0}(D) → ⋯ → HC^{n−1}(B) → HC^{n}(D) → HC^{n}(A) → HC^{n}(B) → …. (3.11)

The proof requires only minor modifications of that of Theorem 3.5 in view of Lemmas 3.6 and 3.7.

4. The existence of long exact sequences of cohomology groups of Banach algebras

As stated above, Wodzicki remarked that his result on the existence of long exact sequences of cyclic and simplicial homology groups can be extended to the continuous case. In Corollary 4 [Wo2] he considered extensions of Banach algebras 0 → B →^{i} A →^{j} D → 0 where B has a left or right bounded approximate identity and stated the existence of long exact sequences associated with the extension under the explicit hypothesis of the admissibility of the extension. Subsequently, he pointed out that this hypothesis is unnecessary. Here we state his result in a new formulation.

4.1 Theorem. Let

0 → B →^{i} A →^{j} D → 0

be an extension of Banach algebras. Suppose B has a left or right bounded approximate identity. Then there exist associated long exact sequences of Banach simplicial homology groups (3.5), of Banach cyclic homology groups (3.10) and of Banach Bar homology groups (3.7), and so HR_{n}(A) = HR_{n}(D) for all n ≥ 0.

Proof. Let us introduce the following notation. Let \underline{n} = (n_{1}, …, n_{l+1}) be an (l + 1)-tuple of integers such that n_{1}, n_{l+1} ≥ 0 and the others are > 0 and let \underline{k} = (k_{1}, …, k_{l}) be an l-tuple of integers such that k_{1} ≥ 0 and the others are > 0. Put |\underline{n}| := n_{1} + ⋯ + n_{l+1} and l(\underline{n}) := l. In view of Lemmas 3.3, 3.4, 3.6 and 3.7, we can see that, up to algebraic isomorphism,

C_{m}(A) = ⨁_{\underline{n},\underline{k}:|\underline{n}|+|\underline{k}|=m+1, l(\underline{n})≥1, l(\underline{k})=l(\underline{n})−1} D^{ˆ⊗n_{1}} ˆ⊗B^{ˆ⊗k_{1}} ˆ⊗ … ˆ⊗D^{ˆ⊗n_{l+1}},

Ker j^{ˆ⊗(m+1)} = ⨁_{\underline{n}, \underline{k}:k_{1}>0, |\underline{n}|+|\underline{k}|=m+1, l(\underline{n})≥1, l(\underline{k})=l(\underline{n})−1} D^{ˆ⊗n_{1}} ˆ⊗B^{ˆ⊗k_{1}} ˆ⊗ … ˆ⊗D^{ˆ⊗n_{l+1}}

and Ker { \atop j^{ˆ⊗(m+1)}} = \{u + Im (id_{A^{ˆ⊗(m+1)}} − t\underline{m}); where u ∈ Ker j^{ˆ⊗(m+1)}\}. Thus the proof is the same as in the algebraic case in Theorem 3.1 [Wo3] and Theorem 3 [Wo2] with filtrations in which algebraic tensor products are replaced by the projective tensor products.

4.2 Theorem. Let

0 → B →^{i} A →^{j} D → 0

be an extension of Banach algebras. Suppose B has a left or right bounded approximate identity. Then there exist associated long exact sequences of Banach simplicial cohomology groups (3.6), of Banach cyclic cohomology groups (3.11) and of Banach Bar cohomology groups (3.8), and so HR^{n}(D) = HR^{n}(A) for all n ≥ 0.

Proof. The existence of the long exact sequences follows from Theorems 4.1, 3.5 and 3.8 and Lemma 3.6.

By Theorem 16 [He2], HR^{n}(D) = {0} for every Banach algebra A with left or right bounded approximate identity and for all n ≥ 0. Hence the final statement follows.

Since C^{∗}-algebras have bounded approximate identities, Theorem 4.2 applies whenever B is a C^{∗}−algebra. By virtue of the main result of [Dix], the Banach algebra K(E) of compact operators on a Banach space E with the bounded compact approximation property has a bounded left approximate identity. For more examples of Banach algebras with a left or right bounded approximate identity see, for example [Pa; Section 5.1].

4.3 Remark. Let us consider the extension of Banach algebras 0 → B →^{i} A →^{j} D → 0, when B has a left or right bounded approximate identity. The existence of the Connes-Tsygan sequence for one of Banach algebras A or D implies the existence of the same exact sequence for the other. This follows from Theorems 15, 16 [He2] and Theorem 4.2.

4.4 Proposition. Let

0 → B →^{i} A →^{j} D → 0,

be an extension of Banach algebras. Then

(i) if B is an amenable Banach algebra, then H^{n}(A) = H^{n}(D) for all n ≥ 2 and there exist exact sequences

0 → D^{tr} → A^{tr} → B^{tr} → H^{1}(D) → H^{1}(A) → 0

and

0 → HC^{2n}(D) → HC^{2n}(A) → B^{tr} → HC^{2n+1}(D) → HC^{2n+1}(A) → 0

for every n ≥ 0; (ii) if B is a C^{∗}-algebra without non-zero bounded traces then

H^{n}(A) = H^{n}(D) and HC^{n}(A) = HC^{n}(D) for all n ≥ 0.

Proof. (i) By assumption B is amenable, and so B has a bounded approximate identity. Hence, by Theorem 4.2, there exist associated long exact sequences of Banach simplicial and cyclic cohomology groups (3.6) and (3.11). Recall that, for an amenable Banach algebra B, H^{n}(B) = {0} for all n ≥ 1 and, by Theorem 25 [He2], HC^{n}(B) is equal to B^{tr} for even n and to 0 for odd n. The result follows in view of Lemma 7.1.32 [BS].

(ii) By Proposition 1.7.3 [Di], any C^{∗}-algebra has a bounded approximate identity and so Theorem 4.2 applies. By Theorem 4.1 and Corollary 3.3 [ChS], for every C^{∗}-algebras without non-zero bounded traces, the Banach simplicial and cyclic cohomology groups vanish for all n ≥ 0.

Note that some results of this proposition were proved by a different approach in [Ly].

4.5 Remark. Let us consider the extension of Banach algebras 0 → B →^{i} A →^{j} D → 0, where B has a left or right bounded approximate identity. Then the condition that D be an amenable Banach algebra implies, by the same arguments as in Theorem 4.4, the following: H^{n}(A) = H^{n}(B) for all n ≥ 1 and there exist exact sequences

0 → D^{tr} → A^{tr} → B^{tr} → 0

and

0 → HC^{2n+1}(A) → HC^{2n+1}(B) → D^{tr} → HC^{2n+2}(A) → HC^{2n+2}(B) → 0

for every n ≥ 0.

As in Theorem 4.4, the condition that D be a C^{∗}-algebra without non-zero bounded traces implies H^{n}(A) = H^{n}(B) and HC^{n}(A) = HC^{n}(B) for all n ≥ 0.

4.6 Examples. Some examples of C^{∗}-algebras without non-zero bounded traces are: (i) The C^{∗}-algebra K(H) of compact operators on an infinite-dimensional Hilbert space H; see [An; Theorem 2]. We can also show that C(Ω, K(H))^{tr} = 0, where Ω is a compact space. (ii) Properly infinite von Neumann algebras U. By Proposition 2.2.4 [Sa] in U there exists a sequence (p_{m}) of mutually orthogonal, equivalent projections with p_{m} ∼ e. Thus, by Theorem 2.1 [Fa], each hermitian element of U is the sum of five commutators. Hence there are no non-zero traces on U. This class includes the C^{∗}-algebra B(H) of all bounded operators on an infinite-dimensional Hilbert space H; see also [Hal] for the statement B(H)^{tr} = 0.

4.7 Examples. Each nuclear C^{∗}-algebra is amenable [Ha]. Some examples of amenable C^{∗}-algebras are: (i) GCR C^{∗}-algebras; in particular, commutative C^{∗}- algebras and the C^{∗}-algebra of compact operators K(H) on a Hilbert space H; (ii) Uniformly hyperfinite algebras (UHF-algebras) [Mu; Remark 6.2.4].

The group algebra L^{1}(G) of Haar integrable functions on an amenable locally compact group G with convolution product is amenable too [Jo1]. For more examples of amenable Banach algebras see, for example [GJW].

4.8 Examples. In [GJW] it is shown that the Banach algebra K(E) of compact operators on a Banach space E with property (𝔸) which was defined in [GJW] is amenable. Property (𝔸) implies that K(E) contains a bounded sequence of projections of unbounded finite rank, and from this it is easy to show (via embedding of matrix algebras) that there is no non-zero bounded trace on K(E). Thus we can see from Theorem 4.4 that, for every extension of Banach algebras

0 → K(E) →^{i} A →^{j} D → 0,

we have

H^{n}(A) = H^{n}(D) and HC^{n}(A) = HC^{n}(D) for all n ≥ 0.

In particular, for the Banach algebra A = B(E) of all bounded operators on a Banach space E, H^{n}(B(E)) = H^{n}(B(E)/K(E)) and HC^{n}(B(E)) = HC^{n}(B(E)/K(E)) for all n ≥ 0. Several classes of Banach spaces have the property (𝔸): l_{p}; 1 < p < ∞; C(K), where K is a compact Hausdorff space; L_{p}(Ω, μ); 1 < p < ∞ (for details and more examples see [Jo1] and [GJW]).

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Zinaida A. Lykova, Department of Mathematics and Statistics, Fylde College, Lancaster University, Lancaster LA1 4YF, England

  E-mail address: maa032@cent1.lancs.ac.uk