RELATIVE COHOMOLOGY OF BANACH ALGEBRAS

ZINAIDA A. LYKOVA

Abstract. Let A be a Banach algebra, not necessarily unital, and let B be a closed subalgebra of A. We establish a connection between the Banach cyclic cohomology group HC^{n}(A) of A and the Banach B-relative cyclic cohomology group HC_{B}^{n}(A) of A. We prove that, for a Banach algebra A with a bounded approximate identity and an amenable closed subalgebra B of A, up to topological isomorphism, HC^{n}(A) = HC_{B}^{n}(A) for all n ≥ 0. We also establish a connection between the Banach simplicial or cyclic cohomology groups of A and those of the quotient algebra A/I by an amenable closed bi-ideal I. The results are applied to the calculation of these groups for certain operator algebras, including von Neumann algebras.

1. Introduction

Much interest has attached in recent years to the computation of cyclic (co)homology groups; see [Lo] for many references. Most of the literature has been devoted to the purely algebraic context, but there have also been papers addressing the calculation of the Banach version of these groups for Banach algebras, and in particular C^{∗}-algebras: see, for example, [ChS1], [Wo] and [He2]. There is an effective tool for computing cyclic (co)homology: the Connes-Tsygan exact sequence, which connects the cyclic (co)homology of many algebras with their simplicial (co)homology. However, it remains the case that these groups can only be calculated for a restricted range of algebras. The purpose of this paper is to describe a technique for the calculation of the Banach simplicial and cyclic cohomology groups of Banach algebras, to establish the basic properties of this technique and to apply it to some natural classes of algebras. The technique involves, for a Banach algebra A, not necessarily unital, the notion of Banach cyclic cohomology groups HC_{B}^{n}(A) of A relative to a closed subalgebra B of A. This concept was introduced and exploited by L. Kadison [Ka1]in the algebraic theory. We establish a connection between the Banach cyclic cohomology group HC^{n}(A) of A and the Banach B-relative cyclic cohomology group HC_{B}^{n}(A) of A. The key result (Theorem 4.1) is that, for a Banach algebra A with a bounded approximate identity and an amenable closed subalgebra B of A, HC^{n}(A) = HC_{B}^{n}(A) for all n ≥ 0. With the aid of this theorem we show, for example, that if R is a von Neumann algebra then, for each n ≥ 0, HC^{n}(R) is the direct sum of the cyclic cohomologies HC^{n}(R_{I_{f}}) and HC^{n}(R_{II_{1}}) of the finite von Neumann algebras R_{I_{f}} and R_{II_{1}} of types I and II appearing in the standard central direct summand decomposition of R (Corollary 4.5). ( In this paper equality of cohomology groups means topological isomorphism of seminormed spaces.)

Key words and phrases. Cohomology of Banach algebras.

I am indebted to the Kansas State University, the University of California at San Diego and the Mathematical Sciences Research Institute for hospitality while this work was carried out. Research at MSRI is supported in part by NSF grant DMS-9022140. It was supported also by an RFFI grant; Project 93-011-156.

In establishing the connection between the Banach cyclic cohomology groups and the Banach relative cyclic cohomology groups of a Banach algebra A, we need to know the connection between the Banach simplicial cohomology groups H^{n}(A, A^{∗}) of A and its relative analogue H_{B}^{n}(A, A^{∗}). Here A^{∗} is the dual Banach space of A. In Section 1 we prove Theorem 1.6 that for a Banach algebra A, an amenable closed subalgebra B of A and a dual A-bimodule M, H^{n}(A, M) = H_{B}^{n}(A, M) for all n ≥ 0. I am grateful to B. E. Johnson for suggesting that this result should be true. He gave a proof in the special case of a Banach algebra A with an identity, an amenable closed subalgebra B of A which contains the identity of A and a unital dual A-bimodule M. He wrote that it should not be too difficult to prove the general case by known techniques (personal communication, 1993). As it transpired, the identical result (for the special case) had been proved by different methods by L. Kadison [Ka2].

E. Christensen and A. M. Sinclair in [ChS2] used the same version of relative Hochschild cohomology group for the computation of the Hochschild cohomology groups of von Neumann algebras. See also [SiSm].

In Section 3 we introduce the Banach relative cyclic cohomology of a Banach algebra and show that the relative Connes-Tsygan exact sequence exists for a Banach algebra with a bounded approximate identity. We do this using ideas of A. Ya. Helemskii [He2]. In this section we also show the existence of morphisms of certain Connes-Tsygan exact sequences.

The results of Section 1 allow us to find, in Section 2, a connection between the cohomology groups of a Banach algebra A and that of a quotient algebra A/I by an amenable closed bi-ideal I. We prove Theorem 2.1 that in this case, for a dual A/I-bimodule M, H^{n}(A, M) = H^{n}(A/I, M) for all n ≥ 0. Thus we can obtain, in Theorem 2.4, the following information about the Banach simplicial cohomology groups: H^{n}(A, A^{∗}) = H^{n}(A/I, (A/I)^{∗}) for all n ≥ 2 and amenable I. A connection between the Banach cyclic cohomology groups of a Banach algebra A and that of a quotient algebra A/I by an amenable closed bi-ideal I is given in Theorem 4.2 of Section 4.

Finally, I would like to thank J. H. Anderson and N. J. Young for some helpful comments.

2. Definitions and notation

Let A be a Banach algebra, not necessarily unital, and let A_{+} be the unitization of A. We denote by e_{+} the adjoined identity and by e an identity of A when it exists.

We recall some notation and terminology used in the homological theory of Banach algebras. Let A be a Banach algebra, not necessarily unital, and let X be a Banach A-bimodule. We define an n-cochain to be a bounded n-linear operator of A×⋯×A into X and we denote the space of n-cochains by C^{n}(A, X). For n = 0 the space C^{0}(A, X) is defined to be X. Let us consider the standard cohomological complex

0 → C^{0}(A, X) →^{δ^{0}} ⋯ → C^{n}(A, X) →^{δ^{n}} C^{n+1}(A, X) → …, (C(A, X))

where the coboundary operator δ^{n} is defined by

(δ^{n}f)(a_{1}, ..., a_{n+1}) = a_{1} · f(a_{2}, ..., a_{n+1})+

∑_{i=1}^{n}(−1)^{i}f(a_{1}, ..., a_{i}a_{i+1}, ..., a_{n+1}) + (−1)^{n+1}f(a_{1}, ..., a_{n}) · a_{n+1}.

The kernel of δ^{n} in C^{n}(A, X) is denoted by Z^{n}(A, X) and its elements are called n−cocycles. The image of δ^{n−1} in C^{n}(A, X) (n ≥ 1) is denoted by N^{n}(A, X) and its elements are called n−coboundaries. An easy computation yields δ^{n+1} ∘ δ^{n} = 0, n ≥ 0.

0.1 Definition. The nth cohomology group of C(A, X) is called the n-dimensional Banach cohomology group of A with coefficients in X. It is denoted by H^{n}(A, X).

Thus H^{n}(A, X) = Z^{n}(A, X)/N^{n}(A, X); it is a complete seminormed space.

Recall that a Banach A-bimodule M = (M_{∗})^{∗}, where M_{∗} is a Banach A-bimodule, is called dual. A Banach algebra A such that H^{1}(A, M) = {0} for all dual A-bimodules M is called amenable.

Now let S be a closed subalgebra of A_{+}. We denote by C_{S}^{n}(A, X) the closed subspace of C^{n}(A, X) of n-cochains ρ such that

ρ(sa_{1}, a_{2}, ..., a_{n}) = s · ρ(a_{1}, ..., a_{n}),
ρ(a_{1}, ..., a_{i−1}, a_{i}s, a_{i+1}, a_{i+2}, ..., a_{n}) = ρ(a_{1}, ..., a_{i−1}, a_{i}, sa_{i+1}, a_{i+2}, ..., a_{n})

and

ρ(a_{1}, a_{2}, ..., a_{n}s) = ρ(a_{1}, ..., a_{n}) · s

for all a_{1}, a_{2}, ..., a_{n} ∈ A, s ∈ S and 1 ≤ i ≤ n. These cochains we shall call S-relative n-cochains. For n = 0 the space C_{S}^{0}(A, X) is defined to be Cen_{S}X =^{def} {x ∈ X : s · x = x · s for all s ∈ S}.

Note that, for each ρ ∈ C_{S}^{n}(A, X), δ^{n}ρ is also an S-relative cochain. Therefore there is a subcomplex in C(A, X) formed by the spaces C_{S}^{n}(A, X). We denote this subcomplex by C_{S}(A, X). The kernel of δ^{n} in C_{S}^{n}(A, X) is denoted by Z_{S}^{n}(A, X) and its elements are called S-relative n−cocycles. The image of δ^{n−1} : C_{S}^{n−1}(A, X) → C_{S}^{n}(A, X) (n ≥ 1) is denoted by N_{S}^{n}(A, X) and its elements are called S-relative n−coboundaries.

0.2 Definition. The nth cohomology group of C_{S}(A, X) is called the n-dimen-sional Banach S-relative cohomology group of A with coefficients in X. It is denoted by H_{S}^{n}(A, X).

When S = ℂ_{+} the subscript S is unnecessary and we omit it.

Throughout the paper id denotes the identity operator. We denote the projective tensor product of Banach spaces by ˆ⊗ and the projective tensor product of left and right Banach A-bimodules by ˆ⊗_{A} [Rie].

3. Relative cohomology of Banach algebras

We need a strengthening of Theorem 4.1 of [JKR] to prove the isomorphism of the cohomology and the relative cohomology of a Banach algebra A for dual A-bimodules.

1.1 Proposition. Let A be a Banach algebra, let B be an amenable closed subalgebra of A, let M be a dual A-bimodule and let n ≥ 1. Suppose ρ ∈ C^{n}(A, M) is such that

(δ^{n}ρ)(a_{1}, ..., a_{n+1}) = 0

if any one of a_{1}, ..., a_{n+1} lies in B. Then there exists ξ ∈ C^{n−1}(A, M) such that

(ρ − δ^{n−1}ξ)(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{n} lies in B.

The proof is the same as that of Theorem 4.1 of [JKR]. □ The following is essentially Lemma 4.1 of [Rin], but with a weakening of the hy-

pothesis.

1.2 Lemma. Let A be a Banach algebra, let B be a closed subalgebra of A, let X be a Banach A-bimodule and let n ≥ 1. Suppose ρ ∈ C^{n}(A, X) is such that

(δ^{n}ρ)(a_{1}, ..., a_{n+1}) = 0

if any one of a_{1}, ..., a_{n+1} lies in B and

ρ(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{n} lies in B. Then ρ ∈ C_{B}^{n}(A, X).

The proof is the same as that of Lemma 4.1 of [Rin]. □ 1.3 Corollary. Let A be a Banach algebra, let B be an amenable closed subalgebra

of A, let M be a dual A-bimodule and let n ≥ 1. Suppose ρ ∈ Z^{n}(A, M). Then there exists ξ ∈ C^{n−1}(A, M) such that

(ρ − δ^{n−1}ξ)(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{n} lies in B. Moreover (ρ − δ^{n−1}ξ) ∈ Z_{B}^{n}(A, M).

1.4 Proposition. Let A be a Banach algebra, let B be a closed subalgebra of A with a bounded approximate identity e_{ν}, ν ∈ Λ, let M = (M_{∗})^{∗} be a dual A-bimodule and let n ≥ 1. Suppose ρ ∈ C_{B}^{n}(A, M) is such that

(δ^{n}ρ)(a_{1}, ..., a_{n+1}) = 0  (3.1)

if any one of a_{1}, ..., a_{n+1} lies in B. Then there exists ξ_{B} ∈ C_{B}^{n−1}(A, M) such that

(ρ − δ^{n−1}ξ_{B})(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{n} lies in B.

Proof. For n = 1, by assumption, for each b ∈ B and ν ∈ Λ,

0 = (δ^{1}ρ)(e_{ν}, b) = e_{ν} · ρ(b) − ρ(e_{ν}b) + ρ(e_{ν}) · b = ρ(e_{ν}) · b,

since ρ ∈ C_{B}^{1}(A, M) and e_{ν} ∈ B. So we obtain

ρ(b) = lim_{ν} ρ(e_{ν}b) = lim_{ν} ρ(e_{ν}) · b = 0.

Hence we can take ξ_{B} = 0.

For n > 1, we construct, inductively on k, ξ_{1}, ..., ξ_{k} in C_{B}^{n−1}(A, M) such that

(ρ − δ^{n−1}ξ_{k})(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{k} lies in B for 1 ≤ k ≤ n. The conclusion of the proposition then follows, with ξ_{B} = ξ_{n}.

To construct ξ_{1}, we consider g_{ν} ∈ C^{n−1}(A, M) given by

g_{ν}(a_{1}, ..., a_{n−1}) = ρ(e_{ν}, a_{1}, ..., a_{n−1}),

for a_{1}, ..., a_{n−1} ∈ A and ν ∈ Λ. Since the Banach space C^{n−1}(A, M) is the dual space of A ˆ⊗ … ˆ⊗A ˆ⊗M_{∗}, for the bounded net g_{ν}, ν ∈ Λ, there exists a subnet g_{μ}, μ ∈ Λ′, which weak^{∗} converges to some cochain g ∈ C^{n−1}(A, M). It is routine to check that g ∈ C_{B}^{n−1}(A, M).

By assumption, for each b ∈ B and ν ∈ Λ,

0 = δ^{n}ρ(e_{ν}, b, a_{2}, ..., a_{n}) =
e_{ν} · ρ(b, a_{2}, ..., a_{n}) − ρ(e_{ν}b, a_{2}, ..., a_{n}) + ρ(e_{ν}, ba_{2}, ..., a_{n})

+ ∑_{i=2}^{n−1}(−1)^{i+1}ρ(e_{ν}, b, a_{2}, ..., a_{i}a_{i+1}, ..., a_{n}) + (−1)^{n+1}ρ(e_{ν}, b, a_{2}, , ..., a_{n−1}) · a_{n} =(3.2)

ρ(e_{ν}, ba_{2}, ..., a_{n})

+ ∑_{i=2}^{n−1}(−1)^{i+1}ρ(e_{ν}, b, a_{2}, ..., a_{i}a_{i+1}, ..., a_{n}) + (−1)^{n+1}ρ(e_{ν}, b, a_{2}, , ..., a_{n−1}) · a_{n}.

Thus

ρ(b, a_{2}, ..., a_{n}) − δ^{n−1}g(b, a_{2}, ..., a_{n}) =
ρ(b, a_{2}, ..., a_{n}) − b · g(a_{2}, ..., a_{n}) + g(ba_{2}, ..., a_{n})

+ ∑_{i=2}^{n−1}(−1)^{i+1}g(b, a_{2}, ..., a_{i}a_{i+1}, ..., a_{n}) + (−1)^{n+1}g(b, a_{2}, , ..., a_{n−1}) · a_{n} =

ρ(b, a_{2}, ..., a_{n}) − lim_{μ} b · ρ(e_{μ}, a_{2}, ..., a_{n}) + lim_{μ}[ρ(e_{μ}, ba_{2}, ..., a_{n})

+ ∑_{i=2}^{n−1}(−1)^{i+1}ρ(e_{μ}, b, a_{2}, ..., a_{i}a_{i+1}, ..., a_{n}) + (−1)^{n+1}ρ(e_{μ}, b, a_{2}, , ..., a_{n−1}) · a_{n}].

The first two terms cancel, and the remaining ones add up to zero by (3.2). This proves the existence of a suitable cochain ξ_{1} = g.

Suppose now that 1 ≤ k < n, and a suitable cochain ξ_{k} ∈ C_{B}^{n−1}(A, M) has been constructed. With ρ − δ^{n−1}ξ_{k} ∈ C_{B}^{n}(A, M) denoted by σ,

σ(a_{1}, ..., a_{n}) = 0  (3.3)

if any one of a_{1}, ..., a_{k} lies in B. In order to continue the inductive process (and so complete the proof of the theorem), it suffices to construct ζ in C_{B}^{n−1}(A, M) such that σ − δ^{n−1}ζ vanishes whenever any one of its first k + 1 arguments lies in B. For then we have ρ − δ^{n−1}(ξ_{k} + ζ) = σ − δ^{n−1}ζ, and we may take ξ_{k+1} = ξ_{k} + ζ. To this end, we consider g_{ν} ∈ C^{n−1}(A, M) given by

g_{ν}(a_{1}, ..., a_{n−1}) = σ(a_{1}, …, a_{k}, e_{ν}, a_{k+1}, …, a_{n−1}),

for a_{1}, ..., a_{n−1} ∈ A and ν ∈ Λ. By the same arguments as in the case k = 1, there exists a subnet g_{μ}, μ ∈ Λ′, which weak^{∗} converges to some cochain g ∈ C^{n−1}(A, M). It can be checked that g ∈ C_{B}^{n−1}(A, M) and

g(a_{1}, ..., a_{n−1}) = 0  (3.4)

if any one of a_{1}, ..., a_{k} lies in B.

In view of the assumption (3.1), for any b ∈ B and ν ∈ Λ,

δ^{n}σ(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n})

= (δ^{n}ρ − (δ^{n} ∘ δ^{n−1})(ξ_{k}))(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n}) = 0.

Hence by the coboundary formula,

0 = δ^{n}σ(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n}) =
a_{1} · σ(a_{2}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n})

+ ∑_{i=1}^{k−1}(−1)^{i}σ(a_{1}, …, a_{i}a_{i+1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n})

+(−1)^{k}σ(a_{1}, …, a_{k}e_{ν}, b, a_{k+2}, …, a_{n}) + (−1)^{k+1}σ(a_{1}, …, a_{k}, e_{ν}b, a_{k+2}, …, a_{n})

+(−1)^{k+2}σ(a_{1}, …, a_{k}, e_{ν}, ba_{k+2}, …, a_{n})

+ ∑_{i=k+2}^{n−1}(−1)^{i+1}σ(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{i}a_{i+1}, …, a_{n})

+(−1)^{n+1}σ(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n−1}) · a_{n}.

By the inductive hypothesis (3.3) the first two terms vanish, and since σ ∈ C_{B}^{n}(A, M), the third and fourth cancel. Thus

0 = (−1)^{k+2}σ(a_{1}, …, a_{k}, e_{ν}, ba_{k+2}, …, a_{n})

+ ∑_{i=k+2}^{n−1}(−1)^{i+1}σ(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{i}a_{i+1}, …, a_{n})  (3.5)

+(−1)^{n+1}σ(a_{1}, …, a_{k}, e_{ν}, b, a_{k+2}, …, a_{n−1}) · a_{n}.

Now consider

(σ − (−1)^{k}δ^{n−1}g)(a_{1}, …, a_{k}, b, a_{k+2}, …, a_{n}) =

σ(a_{1}, …, a_{k}, b, a_{k+2}, …, a_{n})
+(−1)^{k+1}a_{1} · g(a_{2}, …, a_{k}, b, a_{k+2}, …, a_{n})

+ ∑_{i=1}^{k−1}(−1)^{i+k+1}g(a_{1}, …, a_{i}a_{i+1}, …, a_{k}, b, a_{k+2}, …, a_{n})

−g(a_{1}, …, a_{k}b, a_{k+2}, …, a_{n}) + g(a_{1}, …, a_{k}, ba_{k+2}, …, a_{n})

+ ∑_{i=k+2}^{n−1}(−1)^{i+k+1}g(a_{1}, …, a_{k}, b, a_{k+2}, …, a_{i}a_{i+1}, ..., a_{n})

+(−1)^{n+k+1}g(a_{1}, …, a_{k}, b, a_{k+2}, …, a_{n−1}) · a_{n}.

By (3.4), the second and third terms vanish. So, by definition of g, we have

(σ − (−1)^{k}δ^{n−1}g)(a_{1}, …, a_{k}, b, a_{k+2}, …, a_{n}) =

σ(a_{1}, …, a_{k}, b, a_{k+2}, …, a_{n}) − lim_{μ} σ(a_{1}, …, a_{k}b, e_{μ}, a_{k+2}, …, a_{n})

+(−1)^{k} lim_{μ}[(−1)^{k+2}σ(a_{1}, …, a_{k}, e_{μ}, ba_{k+2}, …, a_{n})

+ ∑_{i=k+2}^{n−1}(−1)^{i+1}σ(a_{1}, …, a_{k}, e_{μ}, b, a_{k+2}, …, a_{i}a_{i+1}, …, a_{n})

+(−1)^{n+1}σ(a_{1}, …, a_{k}, e_{μ}, b, a_{k+2}, …, a_{n−1}) · a_{n}].

The first two terms cancel, and the remaining ones add up to zero by (3.5). This shows that, if ζ = (−1)^{k}g, then σ − δ^{n−1}ζ vanishes when its (k + 1)th argument lies in B. When a_{i} = b ∈ B for some i, 1 ≤ i ≤ k, by the inductive hypothesis (3.3) and (3.4), we obtain

(σ − (−1)^{k}δ^{n−1}g)(a_{1}, …, b, …, a_{n}) =

(−1)^{k+i}g(a_{1}, …, a_{i−1}b, a_{i+1}, …, a_{n}) + (−1)^{k+i+1}g(a_{1}, …, a_{i−1}, ba_{i+1}, …, a_{n}) = 0, since g ∈ C_{B}^{n−1}(A, M). Thus σ − δ^{n−1}ζ vanishes when any of its first k + 1 arguments lies in B. As noted above, this completes the proof of the proposition. □

1.5 Proposition. Let A be a Banach algebra, let B be an amenable closed subalgebra of A, let M be a dual A-bimodule and let n ≥ 1. Suppose ρ ∈ C_{B}^{n}(A, M) ∩ N^{n}(A, M). Then there exists ξ ∈ C_{B}^{n−1}(A, M) such that

δ^{n−1}ξ = ρ.

Proof. For n = 1, by Proposition 1.4, there exists ξ_{B} ∈ C_{B}^{0}(A, M) = Cen_{B}M such that

(ρ − δ^{0}ξ_{B})(a_{1}) = 0

if a_{1} ∈ B. By assumption, there exists ξ_{1} ∈ C^{0}(A, M) = M such that ρ = δ^{0}ξ_{1}. Hence

(ρ − δ^{0}ξ_{B})(a_{1}) = (δ^{0}ξ_{1} − δ^{0}ξ_{B})(a_{1}) = δ^{0}(ξ_{1} − ξ_{B})(a_{1}) = a_{1}.(ξ_{1} − ξ_{B}) − (ξ_{1} − ξ_{B}).a_{1} = 0

if a_{1} ∈ B. This implies ξ_{1} − ξ_{B} ∈ Cen_{B}M , and so ξ_{1} ∈ Cen_{B}M.

For n ≥ 2, by Proposition 1.4, there exists ξ_{B} ∈ C_{B}^{n−1}(A, M) such that

(ρ − δ^{n−1}ξ_{B})(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{n} lies in B, and by Lemma 1.2, ρ_{B} =^{def} (ρ − δ^{n−1}ξ_{B}) ∈ C_{B}^{n}(A, M). By assumption, there exists ξ_{1} ∈ C^{n−1}(A, M) such that ρ = δ^{n−1}ξ_{1}. Hence

ρ_{B} = ρ − δ^{n−1}ξ_{B} = δ^{n−1}ξ_{1} − δ^{n−1}ξ_{B} = δ^{n−1}(ξ_{1} − ξ_{B}).

Further, η =^{def} ξ_{1} − ξ_{B} satisfies the assumption of Proposition 1.1, and so there exists β ∈ C^{n−2}(A, M) such that

η_{B}(a_{1}, ..., a_{n−1}) =^{def} (η − δ^{n−2}β)(a_{1}, ..., a_{n−1}) = 0

if any one of a_{1}, ..., a_{n−1} lies in B. Therefore

ρ_{B} = δ^{n−1}(ξ_{1} − ξ_{B}) = δ^{n−1}η = δ^{n−1}(η_{B} + δ^{n−2}β) = δ^{n−1}η_{B}.

By Lemma 1.2, η_{B} ∈ C_{B}^{n−1}(A, M); this implies

ρ = ρ_{B} + δ^{n−1}ξ_{B} = δ^{n−1}η_{B} + δ^{n−1}ξ_{B} = δ^{n−1}(η_{B} + ξ_{B})

and so, for ξ = η_{B} + ξ_{B} ∈ C_{B}^{n−1}(A, M) , we have δ^{n−1}ξ = ρ. □

As was said in the introduction, the following result is based on a communication of B. E. Johnson.

1.6 Theorem. Let A be a Banach algebra, let B be an amenable closed subalgebra of A and let M be a dual A-bimodule. Then

H^{n}(A, M) = H_{B}^{n}(A, M)

for all n ≥ 0.

Proof. The inclusion morphism of cochain objects C_{B}^{n}(A, M) → C^{n}(A, M) induces a morphism of complexes C_{B}(A, M) → C(A, M) and hence morphisms

F_{n} : H_{B}^{n}(A, M) → H^{n}(A, M)

given, for each ρ ∈ Z_{B}^{n}(A, M), by F_{n}(ρ + N_{B}^{n}(A, M)) = ρ + N^{n}(A, M) (see, for example, [He1; Section 0.5.3]).

For n = 0 we have H_{B}^{0}(A, M) = H^{0}(A, M) = Cen_{A}(M).

In the case where n ≥ 1, the morphism F_{n} is injective by Proposition 1.5 and surjective by Corollary 1.3. Hence, by Lemma 0.5.9 [He1], F_{n} is a topological isomorphism. □

In the particular case when A is a unital C^{∗}-algebra, B is the C^{∗}-algebra generated by an amenable group of unitaries and M is a dual normal A-bimodule, the statement of Theorem 1.6 is given in [SiSm]; see Theorem 3.2.7 [SiSm].

1.7 Proposition. Let A_{i} be a Banach algebra with identity e_{i}, i = 1, ..., m, let A be the Banach algebra direct sum ⨁_{i=1}^{m} A_{i} and let M be a dual A-bimodule. Then the canonical projections from A to A_{i}, i = 1, ..., m, induce a topological isomorphism of Banach spaces C_{B}^{n}(A, M) → ⨁_{i=1}^{m} C^{n}(A_{i}, e_{i}Me_{i}), where n ≥ 0 and B is the Banach subalgebra of A generated by {e_{i}, i = 1, ..., m}. Hence

H^{n}(A, M) = ⨁_{i=1}^{m} H^{n}(A_{i}, e_{i}Me_{i})

for all n ≥ 0.

Here e_{i}Me_{i} = {e_{i} · x · e_{i}, x ∈ M} is a Banach A-bimodule.

Proof. For n = 0 it can be checked that

C_{B}^{0}(A, M) = Cen_{B}(M) = ⨁_{i=1}^{m}(e_{i}Me_{i}) = ⨁_{i=1}^{m} C^{0}(A_{i}, e_{i}Me_{i}).

For n ≥ 1 and ρ ∈ C_{B}^{n}(A, M), we have

ρ(a_{1}, ..., a_{n}) = ρ(∑_{i=1}^{m} a_{1}e_{i}, a_{2}, ..., a_{n}) = ∑_{i=1}^{m} e_{i} · ρ(a_{1}e_{i}, a_{2}e_{i}, ..., a_{n}e_{i}) · e_{i}

for a_{1}, …, a_{n} ∈ A. We define cochain maps from C_{B}^{n}(A, M) to ⨁_{i=1}^{m} C^{n}(A_{i}, e_{i}Me_{i}) and back by

J_{n} : ρ ↦ (ρ_{1}, …, ρ_{m})

where ρ_{i}(a_{1}^{i}, …, a_{n}^{i}) = ρ(a_{1}^{i}, …, a_{n}^{i}) for a_{1}^{i}, …, a_{n}^{i} ∈ A_{i} and for i = 1, …, m, and

G_{n} : (ρ_{1}, …, ρ_{m}) ↦ ρ

is given by ρ(a_{1}, ..., a_{n}) = ∑_{i=1}^{m} ρ_{i}(a_{1}e_{i}, a_{2}e_{i}, ..., a_{n}e_{i}) for a_{1}, …, a_{n} ∈ A.

It is clear that J_{n} ∘ G_{n} = id, G_{n} ∘ J_{n} = id and maps J_{n} and G_{n} are bounded. Thus there is a topological isomorhism of complexes C_{B}(A, M) → ⨁_{i=1}^{m} C(A_{i}, e_{i}Me_{i}), and so

H_{B}^{n}(A, M) = ⨁_{i=1}^{m} H^{n}(A_{i}, e_{i}Me_{i}).

Note that B = ⨁_{i=1}^{m} ℂ_{ℶ} is amenable. Hence, by Theorem 1.6, H^{n}(A, M) = H_{B}^{n}(A, M) for all n ≥ 0. The result now follows directly. □

1.8 Proposition. Let R be a von Neumann algebra, let

R = R_{I_{f}} ⊕ R_{I_{∞}} ⊕ R_{II_{1}} ⊕ R_{II_{∞}} ⊕ R_{III}

be the central direct summand decomposition of R into von Neumann algebras of types I_{f}(finite), I_{∞}, II_{1}, II_{∞}, III with the identity e of R decomposing as e = e_{I_{f}} ⊕ e_{I_{∞}} ⊕ e_{II_{1}} ⊕ e_{II_{∞}} ⊕ e_{III} ([Sa; Section 2.2]), and let M be a dual R-bimodule. Then

(i)

H^{n}(R, M) = H^{n}(R_{I_{f}} , e_{I_{f}} Me_{I_{f}}) ⊕ H^{n}(R_{I_{∞}}, e_{I_{∞}}Me_{I_{∞}})⊕
H^{n}(R_{II_{1}}, e_{II_{1}}Me_{II_{1}}) ⊕ H^{n}(R_{II_{∞}}, e_{II_{∞}}Me_{II_{∞}}) ⊕ H^{n}(R_{III}, e_{III}Me_{III})

for all n ≥ 0; (ii) in particular,

H^{n}(R, R^{∗}) = H^{n}(R_{I_{f}} , R_{I_{f}}^{∗}) ⊕ H^{n}(R_{II_{1}}, R_{II_{1}}^{∗})

for all n ≥ 0.

Proof. Part (i) follows from Proposition 1.7. In part (ii) we apply (i) to M = R^{∗}. By Proposition 2.2.4 [Sa], for a properly infinite von Neumann algebra U there exists a sequence (p_{m}) of mutually orthogonal, equivalent projections in U 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 bounded traces on U. Thus, by virtue of Corollary 3.3 [ChS1], for a von Neumann algebra U of one of the types I_{∞}, II_{∞} or III, the simplicial cohomology groups H^{n}(U, U^{∗}) = {0} for all n ≥ 0. □

1.9 Remark. As for finite von Neumann algebras of type I, by [Sa; Theorem 2.3.2], they are the l_{∞}-direct sum of type I_{m} von Neumann algebras R_{I_{m}}, where m < ∞. By Theorem 2.3.3 [Sa] and results of Section 1.22 [Sa], R_{I_{m}} is ∗-isomorphic to the C^{∗}- tensor product Z ⊗_{min} B(H), where Z is the centre of R_{I_{m}} and dim(H) = m. Hence, by Theorem 7.9 [Jo1], finite von Neumann algebras R_{I_{m}} of type I_{m} are amenable, and so their simplicial cohomology groups vanish H^{n}(R_{I_{m}}, R_{I_{m}}^{∗}) = {0} for all n ≥ 1. It is still not clear to the author whether H^{n}(R_{I_{f}} , R_{I_{f}}^{∗}) = {0} for all n ≥ 2.

Note that the statement of Proposition 1.8 (i) is proved in [SiSm], Corollary 3.3.8, for the case of a dual normal R-bimodule M. However, this result does not apply to M = R^{∗}, since R^{∗} is not in general a normal dual module.

Now let us consider two unital Banach algebras A_{1} and A_{2}, a unital Banach A_{1}−A_{2}- bimodule Y, and the natural triangular matrix algebra

U = [ \begin{matrix}A_{1} & Y \\ 0 & A_{2} \end{matrix} ]

with matrix multiplication and norm

∥ [ \begin{matrix}r_{1} & y \\ 0 & r_{2} \end{matrix} ] ∥ = sup_{∥d_{1}∥≤1,∥d_{2}∥≤1} max\{∥r_{1}d_{1} + y · d_{2}∥, ∥r_{2}d_{2}∥\},

where d_{1} ∈ A_{1}, d_{2} ∈ A_{2}. Let e_{ii}, i = 1, 2, denote the idempotents e_{11} = [ \begin{matrix}e_{A_{1}} & 0 \\ 0 & 0 \end{matrix} ]

and e_{22} = [ \begin{matrix}0 & 0 \\ 0 & e_{A_{2}} \end{matrix} ] respectively. Let us also consider the Banach subalgebra B of

U generated by {e_{ii}, i = 1, 2}.

1.10 Proposition. Let A_{1} and A_{2} be unital Banach algebras, let Y be a unital Banach A_{1} − A_{2}-bimodule, and let

U = [ \begin{matrix}A_{1} & Y \\ 0 & A_{2} \end{matrix} ]

be the natural triangular matrix algebra. Then the two canonical projections from U to A_{1} and A_{2} induce a topological isomorphism of Banach spaces C_{B}^{n}(U, U^{∗}) → ⨁_{i=1}^{2} C^{n}(A_{i}, A_{i}^{∗}), n ≥ 0, and hence

H^{n}(U, U^{∗}) = H^{n}(A_{1}, A_{1}^{∗}) ⊕ H^{n}(A_{2}, A_{2}^{∗})

for all n ≥ 0.

The same assertion in a purely algebraic context was proved by L. Kadison in [Ka1]. A key step in Kadison’s proof is the equality H^{n}(U, U^{∗}) = H_{B}^{n}(U, U^{∗}) for all n ≥ 0, where B is the Banach algebra generated by e_{11} and e_{22}. This equality remains valid in the present context by Theorem 1.6. The rest of the proof is as in [Ka1], with insignificant changes.

4. The connection between the cohomologies of A and A/I

Recall Proposition 5.1 [Jo1] that a quotient algebra of an amenable algebra is amenable, and that an extension of an amenable algebra by an amenable bi-ideal is an amenable algebra (this can also be found in Corollary 35 and Proposition 39 [He3]). The following theorem gives some additional information about the cohomology of Banach algebras A and A/I without the assumption that A be amenable.

2.1 Theorem. Let A be a Banach algebra and let I be a closed two-sided ideal of A. Suppose that I is an amenable Banach algebra and M is a dual A/I-bimodule. Then

H^{n}(A, M) = H^{n}(A/I, M)

for all n ≥ 0.

Proof. For n = 0 we have H^{0}(A, M) = H^{0}(A/I, M) = Cen_{A/I}(M). In the case where n ≥ 1 the inclusion morphism of cochain objects C^{n}(A/I, M) → C^{n}(A, M) induces a morphism of complexes C(A/I, M) → C(A, M) and hence morphisms

L_{n} : H^{n}(A/I, M) → H^{n}(A, M)

given, for each ρ̂ ∈ Z^{n}(A/I, M), by

L_{n}(ρ̂ + N^{n}(A/I, M)) = ρ + N^{n}(A, M),

where ρ(a_{1}, ..., a_{n}) = ρ̂(θ(a_{1}), ..., θ(a_{n})) and θ : A → A/I is the natural epimorphism. It is straight-forward to check that if ρ̂ = δ^{n−1}ξ̂ for some ξ̂ ∈ C^{n−1}(A/I, M) then ρ = δ^{n−1}ξ where ξ(a_{1}, ..., a_{n−1}) = ξ̂(θ(a_{1}), ..., θ(a_{n−1})).

By Corollary 1.3, for η ∈ Z^{n}(A, M), there exists ξ ∈ C^{n−1}(A, M) such that

(η − δ^{n−1}ξ)(a_{1}, ..., a_{n}) = 0

if any one of a_{1}, ..., a_{n} lies in I. We can therefore define

η_{I} =^{def} η − δ^{n−1}ξ and η̂_{I}(a_{1} + I, ..., a_{n} + I) =^{def} η_{I}(a_{1}, ..., a_{n}).

Hence for each η ∈ Z^{n}(A, M) there exists η̂_{I} ∈ Z^{n}(A/I, M) such that

L_{n}(η̂_{I} + N^{n}(A/I, M)) = η + N^{n}(A, M),

and so L_{n} is surjective.

Let L_{n}(ρ̂+N^{n}(A/I, M)) = 0, that is, ρ(a_{1}, ..., a_{n}) = ρ̂(θ(a_{1}), ..., θ(a_{n})) ∈ N^{n}(A, M). This implies that there is β ∈ C^{n−1}(A, M) such that ρ = δ^{n−1}β. Further, β satisfies the assumption of Proposition 1.1, and so there exists α ∈ C^{n−2}(A, M) such that

(β − δ^{n−2}α)(a_{1}, ..., a_{n−1}) = 0

if any one of a_{1}, ..., a_{n−1} lies in I. We can define

β_{I} =^{def} β − δ^{n−2}α

and see that ρ = δ^{n−1}β = δ^{n−1}β_{I}. Therefore ρ̂ = δ^{n−1}β̂_{I}, where

β̂_{I}(a_{1} + I, ..., a_{n} + I) = β_{I}(a_{1}, ..., a_{n}).

This proves the injectivity of L_{n}. Hence, by Lemma 0.5.9 [He1], L_{n} is a topological isomorphism. □

2.2 Proposition. Let A be a Banach algebra and let I be a closed two-sided ideal of A. Suppose that I has a bounded approximate identity. Then H_{I}^{0}(A, I^{∗}) = Cen_{I}I^{∗} and

H_{I}^{n}(A, I^{∗}) = {0}

for all n ≥ 1.

Proof. Let us consider the Banach space C_{I}^{n}(A, I^{∗}) which is isometrically isomorphic to the Banach space _{I}h_{I}(A ˆ⊗_{I} … ˆ⊗_{I}A, I^{∗}) of all Banach I-bimodule morphisms from A ˆ⊗_{I} … ˆ⊗_{I}A into I^{∗}. The latter Banach space is isometricaly isomorphic to Cen_{I}(A ˆ⊗_{I} … ˆ⊗_{I}A ˆ⊗_{I}I)^{∗} by Proposition VII.2.17 [He1]. By virtue of the assumption, I has a bounded approximate identity, and so Proposition II.3.13 [He1] gives us an isomophism of Banach I-bimodules A ˆ⊗_{I} … ˆ⊗_{I}A ˆ⊗_{I}I = I. Therefore, there exists an isometric isomorphism of Banach spaces

F_{n} : C_{I}^{n}(A, I^{∗}) → Cen_{I}I^{∗}

for all n ≥ 0.

Now it is routine to check that the diagram

0 → C_{I}^{0}(A, I^{∗}) →^{δ^{0}} C_{I}^{1}(A, I^{∗}) →^{δ^{1}} … → C_{I}^{n}(A, I^{∗}) →^{δ^{n}} C_{I}^{n+1}(A, I^{∗}) → …
↓ F_{0}  ↓ F_{1}  ↓ F_{n}  ↓ F_{n+1}
0 → Cen_{I}I^{∗} →^{η^{0}} Cen_{I}I^{∗} →^{η^{1}} … → Cen_{I}I^{∗} →^{η^{n}} Cen_{I}I^{∗} → …,

is commutative, where η^{n}(f) = 0 for all even n and η^{n}(f) = f for all odd n. The cohomology of the upper complex is, by definition, H_{I}^{n}(A, I^{∗}). Thus the result now follows directly. □

2.3 Corollary. Let A be a Banach algebra and let I be a closed two-sided ideal of A. Suppose that I is an amenable Banach algebra. Then H^{0}(A, I^{∗}) = Cen_{I}I^{∗} and

H^{n}(A, I^{∗}) = {0}

for all n ≥ 1.

Proof. By Theorem 1.6,

H^{n}(A, I^{∗}) = H_{I}^{n}(A, I^{∗})

for all n ≥ 0. Hence the result follows from Proposition 2.2. □

2.4 Theorem. Let A be a Banach algebra and let I be a closed two-sided ideal of A. Suppose that I is an amenable Banach algebra. Then

(i)

H^{n}(A, A^{∗}) = H^{n}(A/I, (A/I)^{∗})

for all n ≥ 2, and the natural map from H^{1}(A/I, (A/I)^{∗}) into H^{1}(A, A^{∗}) is surjective; (ii) if Cen_{I}I^{∗} = {0} then

H^{n}(A, A^{∗}) = H^{n}(A/I, (A/I)^{∗})

for all n ≥ 0.

Proof. We consider the short exact sequence of Banach A-bimodules

0 ← A/I ←^{j} A ←^{i} I ← 0, (I)

where i and j are the natural embedding and quotient mapping respectively, and its dual complex

0 → (A/I)^{∗} →^{j^{∗}} A^{∗} →^{i^{∗}} I^{∗} → 0. (I^{∗})

By virtue of its amenability, I has a bounded approximate identity and so the complex (I^{∗}) is admissible. Hence, by Corollary III.4.11 [He1], there exists a long exact sequence

0 → H^{0}(A, (A/I)^{∗}) → H^{0}(A, A^{∗}) → H^{0}(A, I^{∗}) → H^{1}(A, (A/I)^{∗}) → H^{1}(A, A^{∗}) →

H^{1}(A, I^{∗}) → H^{2}(A, (A/I)^{∗}) → H^{2}(A, A^{∗}) → H^{2}(A, I^{∗}) → …
⋯ → H^{n−1}(A, I^{∗}) → H^{n}(A, (A/I)^{∗}) → H^{n}(A, A^{∗}) → H^{n}(A, I^{∗}) → ….

Recall that, by Corollary 2.3, H^{n}(A, I^{∗}) = {0} for all n ≥ 1. Thus H^{n}(A, (A/I)^{∗}) = H^{n}(A, A^{∗}) (see Lemma 0.5.9 [He1]) for all n ≥ 2. Therefore, by Theorem 2.1,

H^{n}(A/I, (A/I)^{∗}) = H^{n}(A, (A/I)^{∗}) = H^{n}(A, A^{∗})

for all n ≥ 2. □ Note that Cen_{A}A^{∗} coincides with the space

A^{tr} = {f ∈ A^{∗} : f(ab) = f(ba) for all a, b ∈ A}

of continuous traces on A.

Theorem 2.4 applies whenever I is a nuclear C^{∗}-algebra. Other examples are given by the Banach algebra A = B(E) of all bounded operators on a Banach space E with the property (𝔸), which was defined in [GJW], and the closed ideal I = K(E) of compact operators on E. In this case K(E) is amenable [GJW]. The 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 2.4 (ii) that, for a Banach space E with the property (𝔸), H^{n}(B(E), B(E)^{∗}) = HC^{n}(B(E)/K(E), (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 < ∞, where (Ω, μ) is a measure space (for details and more examples see [Jo1] and [GJW]).

In the case of C^{∗}-algebras, we know that the Banach simplicial cohomology groups vanish for C^{∗}-algebras without non-zero bounded traces [ChS1; Corollary 3.3]. Therefore, for an infinite-dimensional Hilbert space H, we obtain

H^{n}(B(H)/K(H), (B(H)/K(H))^{∗}) = H^{n}(B(H), B(H)^{∗}) = {0}

for all n ≥ 0, since K(H)^{tr} = {0} by [An; Theorem 2] and B(H)^{tr} = {0} by [Hal]. One can also see directly that the Calkin algebra has no non-zero bounded trace, and hence has trivial Banach simplicial cohomology.

Notice that, for every Banach algebra A, the vanishing of H^{n}(A, A^{∗}) for all n ≥ 0 is equivalent to the vanishing of the Banach simplicial homology groups H_{n}(A, A) for all n ≥ 0 [He1, Proposition 2.5.28]. The latter relation was established by M. Wodzicki [Wo] in the case A = B(H). He also proved the vanishing of the (Banach) simplicial homology groups for stable C^{∗}-algebras, i.e., for algebras isomorphic to their C^{∗}-tensor product with K(H) for a separable H, and the vanishing of the algebraic simplicial homology groups for A = B(H)/K(H) and a separable H.

Recall from [Di; Sections 4.2, 4.3] that a C^{∗}-algebra A is called CCR (or liminary) if π(A) = K(H) for each irreducible representation (π, H) of A. A C^{∗}-algebra A is called GCR (or postliminary) if each non-zero quotient of A has a non-zero closed two-sided CCR-ideal. Finally we say that A is NGCR (or antiliminary) if it contains no non-zero closed two-sided CCR-ideal. By [Di; Proposition 4.3.3 and 4.3.6], each C^{∗}-algebra A has a largest closed two-sided GCR-ideal I_{α}, and A/I_{α} is NGCR. The following result allow us to reduce the computation of the simplicial cohomology groups of C^{∗}-algebras to the case of NGCR-algebras.

2.5 Proposition. Let A be a C^{∗}-algebra. Then

H^{n}(A, A^{∗}) = H^{n}(A/I_{α}, (A/I_{α})^{∗})

for all n ≥ 1.

Proof. By [Ha; Corollary 4.2], H^{1}(A, A^{∗}) = {0} for every C^{∗}-algebra A. By Theorem 7.9 [Jo1], I_{α} is amenable. Thus the result directly follows from Theorem 2.4 (i). □

5. The existence of the Connes-Tsygan exact sequence

Let A be a Banach algebra, not necessarily unital, and let S be a closed subalgebra of A_{+}. In this section we introduce the Banach version of the concept of S-relative cyclic cohomology HC_{S}^{n}(A) (compare with [Ka1]). We also show that the S-relative Connes-Tsygan exact sequence exists for every Banach algebra A with a bounded approximate identity. This is accomplished with the aid of ideas from [He2].

When S = ℂ_{+} the subscript S is unnecessary and we omit it.

We denote by C_{S}^{n}(A), n = 0, 1, ..., the Banach space of continuous (n + 1)-linear functionals on A such that

f(sa_{0}, a_{1}, ..., a_{n}) = f(a_{0}, a_{1}, ..., a_{n}s)

and, for j = 0, 1, ..., n − 1,

f(a_{0}, ..., a_{j}s, a_{j+1}, ..., a_{n}) = f(a_{0}, ..., a_{j}, sa_{j+1}, ..., a_{n})

for all s ∈ S and a_{0}, ..., a_{n} ∈ A; these functionals we shall call n-dimensional S-relative cochains. We let

t_{n} : C_{S}^{n}(A) → C_{S}^{n}(A), n = 0, 1, ...,

denote the operator given by

t_{n}f(a_{0}, a_{1}, ..., a_{n}) = (−1)^{n}f(a_{1}, ..., a_{n}, a_{0}),

and we set t_{0} = id. The important point is that t_{n}f is an S-relative cochain since

t_{n}f(sa_{0}, a_{1}, ..., a_{n}) = (−1)^{n}f(a_{1}, ..., a_{n}, sa_{0}) = (−1)^{n}f(a_{1}, ..., a_{n}s, a_{0}) = t_{n}f(a_{0}, ..., a_{n}s)

and the other identities follow just as readily. A cochain f ∈ C_{S}^{n}(A) satisfying t_{n}f = f is called cyclic. We let CC_{S}^{n}(A) denote the closed subspace of C_{S}^{n}(A) formed by the cyclic cochains. In particular,

CC_{S}^{0}(A) = C_{S}^{0}(A) = Cen_{S}A^{∗} = {f ∈ A^{∗} : f(sa) = f(as) for all a ∈ A, s ∈ S}.

From the S-relative cochains we form the standard cohomology complex C̃_{S}(A):

0 → C_{S}^{0}(A) →^{δ^{0}} ⋯ → C_{S}^{n}(A) →^{δ^{n}} C_{S}^{n+1}(A) → …,

where the continuous operator δ^{n} is given by the formula

(δ^{n}f)(a_{0}, a_{1}, ..., a_{n+1}) =

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

One can easily check that δ^{n+1} ∘ δ^{n} is indeed 0 for all n and that each δ^{n}f is again an S-relative cochain. It is not difficult to verify that every δ^{n} sends a cyclic cochain again to a cyclic one. Therefore there is a subcomplex in C̃_{S}(A) formed by the spaces CC_{S}^{n}(A). We denote this subcomplex by C̃C_{S}(A), and its differentials are denoted by

δc^{n} : CC_{S}^{n}(A) → CC_{S}^{n+1}(A).

Note that the complex C̃_{S}(A) is a subcomplex of C̃(A) and C̃C_{S}(A) is a subcomplex of C̃C(A) respectively.

3.1 Definition. The nth cohomology of C̃_{S}(A), denoted by H_{S}^{n}(A), is called the n-dimensional Banach S-relative simplicial, or Hochschild, cohomology group of the Banach algebra A. The nth cohomology of C̃C_{S}(A), denoted by HC_{S}^{n}(A), is called the n-dimensional Banach S-relative cyclic cohomology group of A.

Note that, by definition, δc^{0} = δ^{0}, so that HC_{S}^{0}(A) = H_{S}^{0}(A) coincides with the space A^{tr} = {f ∈ A^{∗} : f(ab) = f(ba) for all a, b ∈ A}.

We define HC_{S}^{−1}(A) to be {0}.

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

Further, we need the following complex C̃R_{S}(A):

0 → C_{S}^{0}(A) →^{δr^{0}} ⋯ → C_{S}^{n}(A) →^{δr^{n}} C_{S}^{n+1}(A) → …,

where the continuous operator δr^{n} is given by the formula

(δr^{n}f)(a_{0}, a_{1}, ..., a_{n+1}) = ∑_{i=0}^{n}(−1)^{i}f(a_{0}, ..., a_{i}a_{i+1}, ..., a_{n+1}).

The nth cohomology of C̃R_{S}(A) is denoted by HR_{S}^{n}(A).

Following [He2] we consider the sequence

0 → C̃C_{S}(A) →^{ī} C̃_{S}(A) →^{M̄} C̃R_{S}(A) →^{N̄} C̃C_{S}(A) → 0

of complexes in the category of Banach spaces and continuous operators, where ī denotes the natural inclusion

i_{n} : CC_{S}^{n}(A) → C_{S}^{n}(A),
M_{n} = id − t_{n} : C_{S}^{n}(A) → C_{S}^{n}(A)

and

N_{n} = id + t_{n} + ⋯ + t_{n}^{n} : C_{S}^{n}(A) → CC_{S}^{n}(A).

3.3 Proposition. Let A be a Banach algebra and let S be a closed subalgebra of A_{+}. Then the sequence

0 → C̃C_{S}(A) →^{ī} C̃_{S}(A) →^{M̄} C̃R_{S}(A) →^{N̄} C̃C_{S}(A) → 0

is exact.

The proof is the same as that of Proposition 4 of [He2]. □ 3.4 Proposition. Let A be a Banach algebra with a bounded approximate identity

e_{ν}, ν ∈ Λ, and let S be a closed subalgebra of A_{+}. Then HR_{S}^{n}(A) = {0} for all n ≥ 0. Proof. For f ∈ C_{S}^{n}(A) we define g_{ν} ∈ C^{n−1}(A) by

g_{ν}(a_{0}, ..., a_{n−1}) = f(e_{ν}, a_{0}, ..., a_{n−1}),

for a_{0}, ..., a_{n−1} ∈ A and ν ∈ Λ. Since the Banach space C^{n−1}(A) is the dual space of A ˆ⊗ … ˆ⊗A, for the bounded net g_{ν}, ν ∈ Λ, there exists a subnet g_{μ}, μ ∈ Λ′, which weak^{∗} converges to some cochain g ∈ C^{n−1}(A). It can be checked that g ∈ C_{S}^{n−1}(A).

For each f ∈ C_{S}^{n}(A) such that δr^{n}(f) = 0 and for each ν ∈ Λ,

0 = δr^{n}f(e_{ν}, a_{0}, ..., a_{n})
= f(e_{ν}a_{0}, a_{1}, ..., a_{n}) + ∑_{i=0}^{n−1}(−1)^{i+1}f(e_{ν}, a_{0}, ..., a_{i}a_{i+1}, ..., a_{n}).  (5.1)

Thus we obtain

δr^{n−1}g(a_{0}, ..., a_{n}) = ∑_{i=0}^{n−1}(−1)^{i}g(a_{0}, ..., a_{i}a_{i+1}, ..., a_{n})

= lim_{μ} ∑_{i=0}^{n−1}(−1)^{i}f(e_{μ}, a_{0}, ..., a_{i}a_{i+1}, ..., a_{n}) = f(a_{0}, a_{1}, ..., a_{n}),

by (5.1). □

Note that in the case where S = ℂ_{+} it is easy to see that the statement of Proposition 3.4 is true for every Banach algebra with left or right bounded approximate identity. This and other conditions are given in detail in [He2, Section 2].

3.5 Proposition. Let A be a Banach algebra with a bounded approximate identity and let S be a closed subalgebra of A_{+}. Then the S-relative Connes-Tsygan exact sequence for A

⋯ → H_{S}^{n}(A) →^{B^{n}} HC_{S}^{n−1}(A) →^{S^{n}} HC_{S}^{n+1}(A) →^{I^{n+1}} H_{S}^{n+1}(A) →^{B^{n+1}} HC_{S}^{n}(A) → …

exists.

Proof. By Proposition 3.3, there are two short exact sequences of complexes

0 → C̃C_{S}(A) →^{ī} C̃_{S}(A) →^{M̄} C̃S_{S}(A) → 0

and

0 → C̃S_{S}(A) →^{j̄} C̃R_{S}(A) →^{N̄} C̃C_{S}(A) → 0

where C̃S_{S}(A) is the subcomplex Im(M̄) = Ker(N̄) of C̃R_{S}(A). Hence, by [He1, Chapter 0, Section 5.5], there exist two long exact sequences (*)

⋯ → HC_{S}^{n}(A) →^{H^{n}(ī)} H_{S}^{n}(A) →^{H^{n}(M̄)} HS_{S}^{n}(A) →^{ζ^{n}} HC_{S}^{n+1}(A) →^{H^{n+1}(ī)} H_{S}^{n+1}(A) → …

and (**)

⋯ → HS_{S}^{n}(A) →^{H^{n}(j̄)} HR_{S}^{n}(A) →^{H^{n+1}(N̄)} HC_{S}^{n}(A) →^{η^{n}} HS_{S}^{n+1}(A) →^{H^{n+1}(j̄)} HR_{S}^{n+1}(A) → ….

By Proposition 3.4, HR_{S}^{n}(A) = {0} for all n ≥ 0. Thus we can see from (**) and Proposition 8 [He2] that η^{n} is a topological isomorphism, and so that HC_{S}^{n}(A) = HS_{S}^{n+1}(A) for all n ≥ 0. By setting HC_{S}^{n−1}(A) instead of HS_{S}^{n}(A) in (*), we get the required exact sequence, where I^{n} = H^{n}(ī), S^{n} = ζ^{n−1}∘η^{n} and B^{n} = (η^{n})^{−1}∘H^{n}(M̄). □

3.6 Proposition. Let A be a Banach algebra with a bounded approximate identity and let S be a closed subalgebra of A_{+}. Then the inclusion morphism of cochain objects C_{S}^{n}(A, A^{∗}) → C^{n}(A, A^{∗}) induces a morphism of Connes-Tsygan exact sequences for A, that is, a commutative diagram

… → H_{S}^{n}(A) →^{B^{n}} HC_{S}^{n−1}(A) →^{S^{n}} HC_{S}^{n+1}(A) →^{I^{n+1}} H_{S}^{n+1}(A) →^{B^{n+1}} HC_{S}^{n}(A) → …
↓ ↓ ↓ ↓ ↓
… → H^{n}(A) →^{B^{n}} HC^{n−1}(A) →^{S^{n}} HC^{n+1}(A) →^{I^{n+1}} H^{n+1}(A) →^{B^{n+1}} HC^{n}(A) → ….

Proof. Note that the inclusion morphism of cochain objects C_{S}^{n}(A, A^{∗}) → C^{n}(A, A^{∗}) gives morphisms of two pairs short exact sequences of complexes

0 → C̃C_{S}(A) →^{ī} C̃_{S}(A) →^{M̄} C̃S_{S}(A) → 0
↓ ↓ ↓
0 → C̃C(A) →^{ī} C̃(A) →^{M̄} C̃S(A) → 0

and

0 → C̃S_{S}(A) →^{j̄} C̃R_{S}(A) →^{N̄} C̃C_{S}(A) → 0
↓ ↓ ↓
0 → C̃S(A) →^{j̄} C̃R(A) →^{N̄} C̃C(A) → 0.

By the cohomology analogue of Proposition II.4.2 [Ma], a morphism of two short exact sequences of complexes induces a morphism of long exact cohomology sequences. Hence we have two commutative diagrams (*)

… → HC_{S}^{n}(A) → H_{S}^{n}(A) → HS_{S}^{n}(A) → HC_{S}^{n+1}(A) → H_{S}^{n+1}(A) → …
↓ ↓ ↓ ↓ ↓
… → HC^{n}(A) → H^{n}(A) → HS^{n}(A) → HC^{n+1}(A) → H^{n+1}(A) → …

and (**)

… → HS_{S}^{n}(A) → HR_{S}^{n}(A) → HC_{S}^{n}(A) → HS_{S}^{n+1}(A) → HR_{S}^{n+1}(A) …
↓ ↓ ↓ ↓ ↓
… → HS^{n}(A) → HR^{n}(A) → HC^{n}(A) → HS^{n+1}(A) → HR^{n+1}(A) ….

By Proposition 3.4, HR_{S}^{n}(A) = HR^{n}(A) = {0} for all n ≥ 0. Thus we can see from (**) and Propositon 8 [He2] that there exists a commutative diagram

HC_{S}^{n}(A) = HS_{S}^{n+1}(A)
↓  ↓
HC^{n}(A) = HS^{n+1}(A)

for all n ≥ 0. By setting HC_{S}^{n−1}(A) instead of HS_{S}^{n}(A) and HC^{n−1}(A) instead of HS^{n}(A) in (*), we get the required commutative diagram. □

3.7 Proposition. Let A and D be Banach algebras with right or left bounded approximate identities. Suppose there exists a continuous homomorphism κ : A → D. Then the associated morphism of cochain objects C^{n}(D, D^{∗}) → C^{n}(A, A^{∗}) induces a morphism of Connes-Tzygan exact sequences for A, that is, a commutative diagram

… → H^{n}(D) →^{B^{n}} HC^{n−1}(D) →^{S^{n}} HC^{n+1}(D) →^{I^{n+1}} H^{n+1}(D) →^{B^{n+1}} HC^{n}(D) …
↓ ↓ ↓ ↓ ↓
… → H^{n}(A) →^{B^{n}} HC^{n−1}(A) →^{S^{n}} HC^{n+1}(A) →^{I^{n+1}} H^{n+1}(A) →^{B^{n+1}} HC^{n}(A) ….

The proof requires only minor modifications of that of Proposition 3.6. □ 3.8 Proposition. Let A be a Banach algebra for which one of the following con-

ditions is satisfied:

(i) A has a left or right bounded approximate identity;

(ii) A coincides with the topological square \overline{A^{2}} of A, and that either A is a flat right Banach A-module or ℂ is a flat left Banach A-module.

Then the vanishing of HC^{n}(A) for all n ≥ 0 is equivalent to the vanishing of the Banach cyclic homology HC_{n}(A) for all n ≥ 0.

The definition of flat module can be found in [He1], and the definition of Banach cyclic homology of a Banach algebra can be found, for example, in [He2; Section 5].

Proof. The assumption gives the existence the Connes-Tsygan exact sequence for cohomology of A [He2, Theorems 15, 16]. We can see from this exact sequence that the vanishing of HC^{n}(A) for all n ≥ 0 is equivalent to the vanishing of the simplicial cohomology H^{n}(A, A^{∗}) for all n ≥ 0. The latter relation is equvalent to the vanishing of the simplicial homology H_{n}(A, A) for all n ≥ 0 [He1; Proposition 2.5.28]. As was noted in [He2; Section 5], for the given assumption, the canonical Connes-Tsygan exact sequence for homology of A also exists. Thus it is easy to see from the Connes-Tsygan exact sequence for homology of A that all these relations are equivalent to the vanishing of the cyclic homology HC_{n}(A) for all n ≥ 0. □

6. Relative cyclic cohomology of Banach algebras

4.1 Theorem. Let A be a Banach algebra with a bounded approximate identity, and let B be an amenable closed subalgebra of A. Then

HC^{n}(A) = HC_{B}^{n}(A)

for all n ≥ 0.

Proof. Consider the commutative diagram of Proposition 3.6

0 → HC_{B}^{0}(A) →^{I^{0}} H_{B}^{0}(A) →^{B^{0}} HC_{B}^{−1}(A) →^{S^{0}} HC_{B}^{1}(A) →^{I^{1}} H_{B}^{1}(A) →^{B^{1}} HC_{B}^{0}(A) …
↓ G_{0} ↓ F_{0} ↓ G_{−1} ↓ G_{1} ↓ F_{1} ↓ G_{0}
0 → HC^{0}(A) →^{I^{0}} H^{0}(A) →^{B^{0}} HC^{−1}(A) →^{S^{0}} HC^{1}(A) →^{I^{1}} H^{1}(A) →^{B^{1}} HC^{0}(A) …

… → H_{B}^{n}(A) →^{B^{n}} HC_{B}^{n−1}(A) →^{S^{n}} HC_{B}^{n+1}(A) →^{I^{n+1}} H_{B}^{n+1}(A) →^{B^{n+1}} HC_{B}^{n}(A) → …
↓ F_{n} ↓ G_{n−1} ↓ G_{n+1} ↓ F_{n+1} ↓ G_{n}
… → H^{n}(A) →^{B^{n}} HC^{n−1}(A) →^{S^{n}} HC^{n+1}(A) →^{I^{n+1}} H^{n+1}(A) →^{B^{n+1}} HC^{n}(A) → ….

Note that HC^{0}(A) = HC_{B}^{0}(A) = A^{tr}. By Theorem 1.6,

H^{n}(A) = H^{n}(A, A^{∗}) = H_{B}^{n}(A, A^{∗}) = H_{B}^{n}(A)

for all n ≥ 0.

As an induction hypothesis suppose that the vertical map

G_{i} : HC_{B}^{i}(A) → HC^{i}(A)

is an isomorphism for each i ≤ n. Then it follows from the five lemma [Ma; Lemma 1.3.3] that the middle vertical map G_{n+1} above is an isomorphism. Hence by Lemma 0.5.9 [He1], G_{n+1} is topological isomorphism. □

We note that the assumptions of Theorem 4.1 are obviously satisfied by all C^{∗}- algebras A and all nuclear C^{∗}-subalgebras B of A_{+}. Recall that all nuclear C^{∗}- algebras and only these C^{∗}-algebras are amenable. For example, all GCR-algebras, in particular, all commutative C^{∗}-algebras are amenable. Other examples are given by the Banach algebra A = B(E) and B = K(E), where E is a Banach space with the property (𝔸) (see [GJW]).

We noted in remark after Theorem 2.4 that K(E)^{tr} = {0} for a Banach space with the property (𝔸). Thus one can see from the following theorem that HC^{n}(B(E)) = HC^{n}(B(E)/K(E)) for all n ≥ 0. In particular, for an infinite-dimensional Hilbert space H, we have HC^{n}(B(H)/K(H)) = HC^{n}(B(H)) = {0} for all n ≥ 0 by Theorem 4.1 [ChS1], since (B(H))^{tr} = {0} by [Hal]. Hence, by Proposition 3.8, the Banach cyclic homology of these algebras HC_{n}(B(H)/K(H)) = HC_{n}(B(H)) = {0} for all n ≥ 0.

4.2 Theorem. Let A be a Banach algebra with a right or left bounded approximate identity and let I be a closed two-sided ideal of A. Suppose that I is an amenable Banach algebra. Then

(i) for all even n ≥ 0 the natural map from HC^{n}(A/I) into HC^{n}(A) is injective, and for all odd n ≥ 1 the natural map from HC^{n}(A/I) into HC^{n}(A) is surjective;

(ii) if I^{tr} = {0} (that is, Cen_{I}I^{∗} = {0}) then

HC^{n}(A) = HC^{n}(A/I)

for all n ≥ 0.

Proof. Consider the commutative diagram of Proposition 3.7 for the Banach algebras A and A/I

0 → HC^{0}(A/I) →^{I^{0}} H^{0}(A/I) →^{B^{0}} HC^{−1}(A/I) →^{S^{0}} HC^{1}(A/I) →^{I^{1}} H^{1}(A/I) →^{B^{1}} HC^{0}(A/I)
↓ G_{0} ↓ L_{0} ↓ G_{−1} ↓ G_{1} ↓ L_{1} ↓ G_{0}
0 → HC^{0}(A) →^{I^{0}} H^{0}(A) →^{B^{0}} HC^{−1}(A) →^{S^{0}} HC^{1}(A) →^{I^{1}} H^{1}(A) →^{B^{1}} HC^{0}(A)
… → H^{n}(A/I) →^{B^{n}} HC^{n−1}(A/I) →^{S^{n}} HC^{n+1}(A/I) →^{I^{n+1}} H^{n+1}(A/I) →^{B^{n+1}} HC^{n}(A/I) …
↓ L_{n} ↓ G_{n−1} ↓ G_{n+1} ↓ L_{n+1} ↓ G_{n}
… → H^{n}(A) →^{B^{n}} HC^{n−1}(A) →^{S^{n}} HC^{n+1}(A) →^{I^{n+1}} H^{n+1}(A) →^{B^{n+1}} HC^{n}(A) ….

Note that the maps G_{0} : HC^{0}(A/I) → HC^{0}(A) : f ↦ f ∘ g and L_{0} : H^{0}(A/I) → H^{0}(A) : f ↦ f ∘ g are injective. By Theorem 2.4,

H^{n}(A) = H^{n}(A, A^{∗}) = H^{n}(A/I, (A/I)^{∗}) = H^{n}(A/I),

so that L_{n} is an isomorphism, for all n ≥ 2 and the natural L_{1} map from

H^{1}(A/I, (A/I)^{∗})

into H^{1}(A, A^{∗}) is surjective. Then it follows from the five lemma that the middle vertical map G_{1} above is surjective.

Suppose, inductively, that the vertical map, for each i ≤ n,

G_{i} : HC^{i}(A/I) → HC^{i}(A)

is injective if i is even and surjective if i is odd. The result then follows from the five lemma. □

Note that in particular Theorem 4.2 applies whenever A is a C^{∗}-algebra and I is an amenable closed ideal.

4.3 Proposition. Let A and D be Banach algebras with right or left bounded approximate identities, and let κ : A → D be a continuous homomorphism. If κ induces a topological isomorphism

H^{n}(D, D^{∗}) → H^{n}(A, A^{∗})

for all n ≥ 0, then it induces a topological isomorphism

HC^{n}(D) → HC^{n}(A)

for all n ≥ 0, and conversely.

Proof. The forward implication is a repetition of that of Theorem 4.1 with the commutative diagram of Proposition 3.7. The converse statement follows easily from the five lemma. □

4.4 Proposition. Let A_{i} be a Banach algebra with identity e_{i}, i = 1, ..., m, and let A be the Banach algebra direct sum ⨁_{i=1}^{m} A_{i}. Then

HC^{n}(A) = ⨁_{i=1}^{m} HC^{n}(A_{i})

for all n ≥ 0.

Proof. By Theorem 4.1, HC^{n}(A) = HC_{B}^{n}(A), where n ≥ 0 and B is the Banach subalgebra of A generated by {e_{i}, i = 1, ..., m}. By Proposition 1.7, the canonical projections from A to A_{i}, i = 1, ..., m, induce a topological isomorphism of complexes C_{B}(A, A^{∗}) → ⨁_{i=1}^{m} C(A_{i}, A_{i}^{∗}). Thus HC_{B}^{n}(A) = ⨁_{i=1}^{m} HC^{n}(A_{i}). The result now follows directly. □

4.5 Corollary. Let R be a von Neumann algebra, let

R = R_{I_{f}} ⊕ R_{I_{∞}} ⊕ R_{II_{1}} ⊕ R_{II_{∞}} ⊕ R_{III}

be the central direct summand decomposition of R into von Neumann algebras of types I_{f}, I_{∞}, II_{1}, II_{∞}, III. Then

HC^{n}(R) = HC^{n}(R_{I_{f}}) ⊕ HC^{n}(R_{II_{1}})

for all n ≥ 0.

Proof. As we noted in the proof of Proposition 1.8, there are no non-zero bounded traces on R_{I_{∞}}, R_{II_{∞}} and R_{III}. Thus, by Theorem 4.1 [ChS1], their Banach cyclic cohomology groups vanish for all n ≥ 0. □

Note that, by Theorem 25 [He2], for a von Neumann algebra R_{I_{m}} of type I_{m}, where m < ∞, we have HC^{n}(R_{I_{m}}) = R_{I_{m}}^{tr} for all even n, and HC^{n}(R_{I_{m}}) = {0} for all odd n.

4.6 Corollary. Let A be a C^{∗}-algebra without non-zero bounded traces. Then the Banach cyclic homology HC_{n}(A) = {0} for all n ≥ 0.

Proof. It follows from Proposition 3.8 and Theorem 4.1 [ChS1]. □ Note that in particular Corollary 4.6 applies whenever A is a properly infinite von

Neumann algebra (see Proposition 1.8), or a stable C^{∗}-algebra [Fa; Theorem 1.1]. The calculation of the (Banach) cyclic homology groups for stable C^{∗}-algebras, for B(H) and for the Calkin algebra on a separable Hilbert space was given in [Wo].

Now we give the Banach version of the algebraic statement of Theorem 9 [Ka1] of L. Kadison.

4.7 Proposition. Let A_{1} and A_{2} be unital Banach algebras, let Y be a unital Banach A_{1} − A_{2}-bimodule, and let

U = [ \begin{matrix}A_{1} & Y \\ 0 & A_{2} \end{matrix} ]

the natural triangular matrix algebra. Then the two canonical projections from U to A_{1} and A_{2} induce a topological isomorphism

HC^{n}(U) = HC^{n}(A_{1}) ⊕ HC^{n}(A_{2})

for all n ≥ 0.

Proof. By Theorem 4.1 and Proposition 1.10, we obtain

HC^{n}(U) = HC_{B}^{n}(U) = HC^{n}(A_{1}) ⊕ HC^{n}(A_{2}),

where the amenable subalgebra B of U was defined before Proposition 1.10. □

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

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