CYCLIC COHOMOLOGY OF CERTAIN NUCLEAR FRÉCHET
AND DF ALGEBRAS
ZINAIDA A. LYKOVA
Abstract. We give explicit formulae for the continuous Hochschild and cyclic
homology and cohomology of certain ⊗̂-algebras. We use well-developed homolog-
ical techniques together with some niceties of the theory of locally convex spaces
to generalize the results known in the case of Banach algebras and their inverse
limits to wider classes of topological algebras. To this end we show that, for a con-
tinuous morphism ϕ : X → Y of complexes of complete nuclear DF -spaces, the
isomorphism of cohomology groups Hn(ϕ) : Hn(X ) → Hn(Y) is automatically
topological. The continuous cyclic-type homology and cohomology are described
up to topological isomorphism for the following classes of biprojective ⊗̂-algebras:
the tensor algebra E⊗̂F generated by the duality (E,F, 〈·, ·〉) for nuclear Fréchet
spaces E and F or for nuclear DF -spaces E and F ; nuclear biprojective Köthe
algebras λ(P ) which are Fréchet spaces or DF -spaces; the algebra of distributions
E∗(G) on a compact Lie group G.
2000 Mathematics Subject Classification: Primary 19D55, 22E41, 16E40, 46H40.
1. Introduction
Cyclic cohomology groups of topological algebras play an essential role in non-
commutative geometry [2]. There has been a number of papers addressing the
calculation of cyclic-type continuous homology and cohomology groups of some Ba-
nach, C∗- and topological algebras; see, e.g., [2, 8, 14, 17, 18, 20, 32]. However, it
remains difficult to describe these groups explicitly for many topological algebras.
To compute the continuous Hochschild and cyclic cohomology groups of Fréchet
algebras one has to deal with complexes of complete DF -spaces. Here, in addition
to presenting known homological techniques we also supply technical enhancements
that permit the necessary generalization of results known in the case of Banach
algebras and their inverse limits to wider classes of topological algebras notably to
those that occur in noncommutative geometry.
The category of Banach spaces has the useful property that it is closed under
passage to dual spaces. Fréchet spaces do not have this property: the strong dual
Date: 22 August 2007.
Key words and phrases. Cyclic cohomology, Hochschild cohomology, nuclear DF -spaces, locally
convex algebras, nuclear Fréchet algebra.
I am indebted to the Isaac Newton Institute for Mathematical Sciences at Cambridge for hos-
pitality and for generous financial support from the programme on Noncommutative Geometry
while this work was carried out.
http://arxiv.org/abs/0704.1019v2
2 Z. A. Lykova
of a Fréchet space is a complete DF -space. DF -spaces have the awkward feature
that their closed subspaces need not be DF -spaces. However, closed subspaces of
complete nuclear DF -spaces are again DF -spaces [21, Proposition 5.1.7].
In Section 3 we use the strongest known results on the open mapping theorem to
give sufficient conditions on topological spaces E and F to imply that any continuous
linear operator T from E∗ onto F ∗ is open. This allows us to prove the following
results. In Lemma 3.6 we show that, for a continuous morphism ϕ : X → Y of
complexes of complete nuclear DF -spaces, the isomorphism of cohomology groups
Hn(ϕ) : Hn(X )→ Hn(Y) is automatically topological.
We use this fact to describe explicitly up to topological isomorphism the continuous
Hochschild and cyclic cohomology groups of nuclear ⊗̂-algebras A which are Fréchet
spaces or DF -spaces and have trivial Hochschild homology HHn(A) for all n ≥ 1
(Theorem 5.3). In Proposition 4.4, under the same condition on HHn(A), we give
explicit formulae, up to isomorphism of linear spaces, for continuous cyclic-type
homology of A in a more general category of underlying spaces.
In Theorem 6.8 the continuous cyclic-type homology and cohomology groups are
described up to topological isomorphism for the following classes of biprojective ⊗̂-
algebras: the tensor algebra E⊗̂F generated by the duality (E, F, 〈·, ·〉) for nuclear
Fréchet spaces or for nuclear complete DF -spaces E and F ; nuclear biprojective
Fréchet Köthe algebras λ(P ); nuclear biprojective Köthe algebras λ(P )∗ which are
DF -spaces; the algebra of distributions E∗(G) and the algebra of smooth functions
E(G) on a compact Lie group G.
2. Definitions and notation
We recall some notation and terminology used in homology and in the theory of
topological algebras. Homological theory can be found in any relevant textbook, for
instance, Loday [16] for the pure algebraic case and Helemskii [7] for the continuous
case.
Throughout the paper ⊗̂ is the projective tensor product of complete locally
convex spaces, by X⊗̂n we mean the n-fold projective tensor power X⊗̂ . . . ⊗̂X of
X , and id denotes the identity operator.
We use the notation Ban, Fr and LCS for the categories whose objects are Banach
spaces, Fréchet spaces and complete Hausdorff locally convex spaces respectively,
and whose morphisms in all cases are continuous linear operators. For topological
homology theory it is important to find a suitable category for the underlying spaces
of the algebras and modules. In [7] Helemskii constructed homology theory for the
following categories Φ of underlying spaces, for which he used the notation (Φ, ⊗̂).
Definition 2.1. ([7, Section II.5]) A suitable category for underlying spaces of
the algebras and modules is an arbitrary complete subcategory Φ of LCS having the
following properties:
(i) if Φ contains a space, it also contains all those spaces topologically isomorphic
to it;
Cyclic cohomology of nuclear Fréchet and DF algebras 3
(ii) if Φ contains a space, it also contains any of its closed subspaces and the
completion of any its Hausdorff quotient spaces;
(iii) Φ contains the direct sum and the projective tensor product of any pair of its
spaces;
(iv) Φ contains C.
Besides Ban, Fr and LCS important examples of suitable categories Φ are the
categories of complete nuclear spaces [31, Proposition 50.1], nuclear Fréchet spaces
and complete nuclear DF -spaces. As to the above properties for the category of
complete nuclear DF -spaces, recall the following results. By [12, Theorem 15.6.2],
if E and F are complete DF -spaces, then E⊗̂F is a complete DF -space. By [21,
Proposition 5.1.7], a closed linear subspace of a complete nuclear DF -space is also
a complete nuclear DF -space. By [21, Proposition 5.1.8], each quotient space of a
complete nuclear DF -space by a closed linear subspace is also a complete nuclear
DF -space.
By definition a ⊗̂-algebra is a complete Hausdorff locally convex algebra with
jointly continuous multiplication. A left ⊗̂-module X over a ⊗̂-algebra A is a com-
plete Hausdorff locally convex spaceX together with the structure of a leftA-module
such that the map A×X → X , (a, x) 7→ a ·x is jointly continuous. For a ⊗̂-algebra
A, ⊗̂A is the projective tensor product of left and right A-⊗̂-modules (see [6], [7,
II.4.1]). The category of left A-⊗̂-modules is denoted by A-mod and the category
of A-⊗̂-bimodules is denoted by A-mod-A.
Let K be one of the above categories. A chain complex X∼ in the category K is a
sequence of Xn ∈ K and morphisms dn
· · · ← Xn
← Xn+1
← Xn+2 ← . . .
such that dn ◦ dn+1 = 0 for every n. The homology groups of X∼ are defined by
Hn(X∼) = Ker dn−1/Im dn.
A continuous morphism of chain complexes ψ∼ : X∼ → P∼ induces a continuous
linear operator Hn(ψ∼) : Hn(X∼)→ Hn(P∼) [9, Definition 0.4.22].
If E is a topological vector space E∗ denotes its dual space of continuous linear
functionals. Throughout the paper, E∗ will always be equipped with the strong
topology unless otherwise stated. The strong topology is defined on E∗ by taking as
a basis of neighbourhoods of 0 the family of polars V 0 of all bounded subsets V of
E; see [31, II.19.2].
For any ⊗̂-algebra A, not necessarily unital, A+ is the ⊗̂-algebra obtained by
adjoining an identity to A. For a ⊗̂-algebra A, the algebra Ae = A+⊗̂A
called the enveloping algebra of A, where A
+ is the opposite algebra of A+ with
multiplication a · b = ba.
A complex of A-⊗̂-modules and their morphisms is called admissible if it splits
as a complex in LCS [7, III.1.11]. A module Y ∈ A-mod is called flat if for any
admissible complex X of right A-⊗̂-modules the complex X⊗̂AY is exact. A module
Y ∈ A-mod-A is called flat if for any admissible complex X of A-⊗̂-bimodules the
4 Z. A. Lykova
complex X⊗̂AeY is exact. For Y,X ∈ A-mod-A, we shall denote by Tor
n (X, Y ) the
nth homology of the complex X⊗̂AeP, where 0← Y ← P is a projective resolution
of Y in A-mod-A, [7, Definition III.4.23].
It is well known that the strong dual of a Fréchet space is a complete DF -space
and that nuclear Fréchet spaces and complete nuclear DF -spaces are reflexive [21,
Theorem 4.4.12]. Moreover, the correspondence E ↔ E∗ establishes a one-to-one
relation between nuclear Fréchet spaces and complete nuclear DF -spaces [21, The-
orem 4.4.13]. DF -spaces were introduced by A. Grothendieck in [5].
Further we shall need the following technical result which extends a result of
Johnson for the Banach case [11, Corollary 1.3].
Proposition 2.2. Let (X , d) be a chain complex of
(a) Fréchet spaces and continuous linear operators, or
(b) complete nuclear DF -spaces and continuous linear operators,
and let N ∈ N. Then the following statements are equivalent:
(i) Hn(X , d) = {0} for all n ≥ N and HN−1(X , d) is Hausdorff;
(ii) Hn(X ∗, d∗) = {0} for all n ≥ N.
Proof. Recall that Hn(X , d) = Ker dn−1/Im dn and H
n(X ∗, d∗) = Ker d∗n/Im d
Let L be the closure of Im dN−1 in XN−1. Consider the following commutative
diagram
0 ← L
←− XN
←− XN+1
←− . . .
↓ i ւ dN−1
in which i is the natural inclusion and j is a corestriction of dN−1. The dual com-
mutative diagram is the following
0 → L∗
−→ X∗N
−→ X∗N+1
−→ . . .
↑ i∗ ր d∗N−1
X∗N−1
It is clear that HN−1(X , d) is Hausdorff if and only if j is surjective. Since i is
injective, condition (i) is equivalent to the exactness of diagram (1). On the other
hand, by the Hahn-Banach theorem, i∗ is surjective. Thus condition (ii) is equivalent
to the exactness of diagram (2).
In the case of Fréchet spaces, by [18, Lemma 2.3], the exactness of the complex
(1) is equivalent to the exactness of the complex (2).
In the case of complete nuclear DF -spaces, by [21, Proposition 5.1.7], L is the
strong dual of a nuclear Fréchet space. By [21, Theorem 4.4.12], complete nuclear
DF -spaces are reflexive, and therefore the complex (1) is the dual of the complex
(2) of nuclear Fréchet spaces and continuous linear operators. By [18, Lemma 2.3],
the exactness of the complex (1) is equivalent to the exactness of the complex (2).
The proposition is proved. �
Cyclic cohomology of nuclear Fréchet and DF algebras 5
3. The open mapping theorem in complete nuclear DF -spaces
It is known that there exist closed linear subspaces of DF -spaces that are not
DF -spaces. For nuclear spaces, however, we have the following.
Lemma 3.1. [21, Proposition 5.1.7] Each closed linear subspace F of the strong dual
E∗ of a nuclear Fréchet space E is also the strong dual of a nuclear Fréchet space.
In a locally convex space a subset is called a barrel if it is absolutely convex,
absorbent and closed. Every locally convex space has a neighbourhood base con-
sisting of barrels. A locally convex space is called a barrelled space if every barrel
is a neighbourhood [26]. By [26, Theorem IV.1.2], every Fréchet space is barrelled.
By [26, Corollary IV.3.1], a Hausdorff locally convex space is reflexive if and only
if it is barrelled and every bounded set is contained in a weakly compact set. Thus
the strong dual of a nuclear Fréchet space is barrelled. For a generalization of the
open mapping theorem to locally convex spaces, V. Pták introduced the notion of
B-completeness in [24]. A subspace Q of E∗ is said to be almost closed if, for each
neighbourhood U of 0 in E, Q∩U0 is closed in the relative weak* topology σ(E∗, E)
on U0. A locally convex space E is said to be B-complete or fully complete if each
almost closed subspace of E∗ is closed in the weak* topology σ(E∗, E).
Theorem 3.2. [24]. Let E be a B-complete locally convex space and F be a barrelled
locally convex space. Then a continuous linear operator f of E onto F is open.
Recall [10, Theorem 4.1.1] that a locally convex space E is B-complete if and only
if each linear continuous and almost open mapping f of E onto any locally convex
space F is open. By [10, Proposition 4.1.3], every Fréchet space is B-complete.
Theorem 3.3. Let E be a semi-reflexive metrizable barrelled space, F be a Hausdorff
reflexive locally convex space and let E∗ and F ∗ be the strong duals of E and F
respectively. Then a continuous linear operator T of E∗ onto F ∗ is open.
Proof. By [10, Theorem 6.5.10] and by [10, Corollary 6.2.1], the strong dual E∗
of a semi-reflexive metrizable barrelled space E is B-complete. By [26, Corollary
IV.3.2], if a Hausdorff locally convex space is reflexive, so is its dual under the strong
topology. By [26, Corollary IV.3.1], a Hausdorff reflexive locally convex space is
barrelled. Hence F ∗ is a barrelled locally convex space. Therefore, by Theorem 3.2,
T is open. �
Corollary 3.4. Let E and F be nuclear Fréchet spaces and let E∗ and F ∗ be the
strong duals of E and F respectively. Then a continuous linear operator T of E∗
onto F ∗ is open.
For a continuous morphism of chain complexes ψ∼ : X∼ → P∼ in Fr, a surjective
map Hn(ϕ) : Hn(X ) → Hn(Y) is automatically open, see [7, Lemma 0.5.9]. To get
the corresponding result for dual complexes of Fréchet spaces one has to assume
nuclearity.
6 Z. A. Lykova
Lemma 3.5. Let (X , dX ) and (Y , dY) be chain complexes of nuclear Fréchet spaces
and continuous linear operators and let (X ∗, d∗X ) and (Y
∗, d∗Y) be their strong dual
complexes. Let ϕ : X ∗ → Y∗ be a continuous morphism of complexes. Suppose that
ϕ∗ = H
n(ϕ) : Hn(X ∗, d∗X )→ H
n(Y∗, d∗Y)
is surjective. Then ϕ∗ is open.
Proof. Let σY∗ : Ker (d
Y)n → H
n(Y∗, d∗Y) be the quotient map. Consider the map
ψ : Ker (d∗X )n ⊕ Y
n−1 → Ker (d
Y)n ⊂ Y
given by (x, y) 7→ ϕn(x) + (d
Y)n−1(y).
By Lemma 3.1, Ker (d∗X )n and Ker (d
Y)n are the strong duals of nuclear Fréchet
spaces and hence are barrelled. By [10, Theorem 6.5.10] and [10, Corollary 6.2.1],
the strong dual of a semi-reflexive metrizable barrelled space is B-complete. Thus
Ker (d∗X )n, Y
n−1 and
Ker (d∗X )n ⊕ Y
∼= [(Ker (d∗X )n)
∗ ⊕ Yn−1]
are B-complete. By assumption ϕ∗ maps H
n(X ∗, d∗X ) onto H
n(Y∗, d∗Y), which im-
plies that ψ is a surjective linear continuous operator from the B-complete locally
convex space Ker (d∗X )n ⊕ Y
n−1 to the barrelled locally convex space Ker (d
Therefore, by Theorem 3.2, ψ is open. Consider the diagram
Ker (d∗X )n ⊕ Y
→ Ker (d∗X )n
→ Hn(X ∗, d∗X )
↓ ψ ↓ ϕ∗
Ker (d∗Y)n
−→ Hn(Y∗, d∗Y)
in which j is a projection onto a direct summand and σX ∗ and σY∗ are the natural
quotient maps. Obviously this diagram is commutative. Note that the projection j
and quotient maps σX ∗ , σY∗ are open. As ψ is also an open map, so is σY∗ ◦ ψ =
ϕ∗ ◦ σX ∗ ◦ j. Since σX ∗ ◦ j is continuous, ϕ∗ is open. �
Corollary 3.6. Let (X , dX ) and (Y , dY) be cochain complexes of complete nuclear
DF -spaces and continuous linear operators, and let ϕ : X → Y be a continuous
morphism of complexes. Suppose that ϕ∗ = H
n(ϕ) : Hn(X , dX ) → H
n(Y , dY) is
surjective. Then ϕ∗ is open.
Proof. By [21, Theorem 4.4.13], (X , dX ) and (Y , dY) are strong duals of chain com-
plexes (X ∗, d∗X ) and (Y
∗, d∗Y) of nuclear Fréchet spaces and continuous operators.
The result follows from Lemma 3.5. �
4. Cyclic and Hochschild cohomology of some ⊗̂-algebras
One can consult the books by Loday [16] or Connes [2] on cyclic-type homological
theory.
Let A be a ⊗̂-algebra and let X be an A-⊗̂-bimodule. We assume here that the
category of underlying spaces Φ has the properties from Definition 2.1. Let us recall
the definition of the standard homological chain complex C∼(A, X). For n ≥ 0, let
Cyclic cohomology of nuclear Fréchet and DF algebras 7
Cn(A, X) denote the projective tensor product X⊗̂A
. The elements of Cn(A, X)
are called n-chains. Let the differential dn : Cn+1 → Cn be given by
dn(x⊗ a1 ⊗ . . .⊗ an+1) = x · a1 ⊗ . . .⊗ an+1+
(−1)k(x⊗ a1 ⊗ . . .⊗ akak+1 ⊗ . . .⊗ an+1) + (−1)
n+1(an+1 · x⊗ a1 ⊗ . . .⊗ an)
with d−1 the null map. The homology groups of this complex Hn(C∼(A, X)) are
called the continuous Hochschild homology groups of A with coefficients in X and
denoted by Hn(A, X) [7, Definition II.5.28]. We also consider the cohomology
groups Hn((C∼(A, X))
∗) of the dual complex (C∼(A, X))
∗ with the strong dual
topology. For Banach algebras A, Hn((C∼(A, X))
∗) is topologically isomorphic to
the Hochschild cohomologyHn(A, X∗) of A with coefficients in the dual A-bimodule
X∗ [7, Definition I.3.2 and Proposition II.5.27]. The weak bidimension of a Fréchet
algebra A is
dbwA = inf{n : H
n+1(C∼(A, X)
∗) = {0} for all Fréchet A−bimodules X}.
The continuous bar and ‘naive’ Hochschild homology of a ⊗̂-algebra A are defined
respectively as
Hbar∗ (A) = H∗(C(A), b
′) and Hnaive∗ (A) = H∗(C(A), b),
where Cn(A) = A
⊗̂(n+1), and the differentials b, b′ are given by
b′(a0 ⊗ · · · ⊗ an) =
(−1)i(a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an) and
b(a0 ⊗ · · · ⊗ an) = b
′(a0 ⊗ · · · ⊗ an) + (−1)
n(ana0 ⊗ · · · ⊗ an−1).
Note that Hnaive∗ (A) is just another way of writing H∗(A,A), the continuous homol-
ogy of A with coefficients in A, as described in [7, 11].
There is a powerful method based on mixed complexes for the study of the cyclic-
type homology groups; see papers by C. Kassel [13], J. Cuntz and D. Quillen [4] and
J. Cuntz [3]. We shall present this method for the category LCS of locally convex
spaces and continuous linear operators; see [1] for the category of Fréchet spaces. A
mixed complex (M, b, B) in the category LCS is a familyM = {Mn}n≥0 of locally
convex spaces Mn equipped with continuous linear operators bn : Mn → Mn−1 and
Bn : Mn → Mn+1, which satisfy the identities b
2 = bB + Bb = B2 = 0. We assume
that in degree zero the differential b is identically equal to zero. We arrange the
8 Z. A. Lykova
mixed complex (M, b, B) in the double complex
. . . . . . . . . . . .
b ↓ b ↓ b ↓
b ↓ b ↓
There are three types of homology theory that can be naturally associated with a
mixed complex. The Hochschild homology Hb∗(M) of (M, b, B) is the homology of
the chain complex (M, b), that is,
Hbn(M) = Hn(M, b) = Ker {bn :Mn →Mn−1}/Im {bn+1 :Mn+1 →Mn}.
To define the cyclic homology of (M, b, B), let us denote by BcM the total complex
of the above double complex, that is,
· · · → (BcM)n
→ (BcM)n−1 → · · ·
→ (BcM)0 → 0,
where the spaces
(BcM)0 =M0, . . . , (BcM)2k−1 =M1 ⊕M3 ⊕ · · · ⊕M2k−1
(BcM)2k =M0 ⊕M2 ⊕ · · · ⊕M2k
are equipped with the product topology, and the continuous linear operators b+B
are defined by
(b+B)(y0, . . . , y2k) = (by2 +By0, . . . , by2k +By2k−2)
(b+B)(y1, . . . , y2k+1) = (by1, . . . , by2k+1 +By2k−1).
The cyclic homology of (M, b, B) is defined to be H∗(BcM, b+B). It is denoted by
Hc∗(M, b, B).
The periodic cyclic homology of (M, b, B) is defined in terms of the complex
· · · → (BpM)ev
→ (BpM)odd
→ (BpM)ev
→ (BpM)odd → . . . ,
where even/odd chains are elements of the product spaces
(BpM)ev =
M2n and (BpM)odd =
M2n+1,
respectively. The spaces (BpM)ev/odd are locally convex spaces with respect to the
product topology [15, Section 18.3.(5)]. The continuous differential b+B is defined
as an obvious extension of the above. The periodic cyclic homology of (M, b, B) is
Hpν (M, b, B) = Hν(BpM, b+B), where ν ∈ Z/2Z.
There are also three types of cyclic cohomology theory associated with the mixed
complex, obtained when one replaces the chain complex of locally convex spaces
Cyclic cohomology of nuclear Fréchet and DF algebras 9
by its dual complex of strong dual spaces. For example, the cyclic cohomology
associated with the mixed complex (M, b, B) is defined to be the cohomology of
the dual complex ((BcM)
∗, b∗ + B∗) of strong dual spaces and dual operators; it is
denoted by H∗c (M
∗, b∗, B∗).
Consider the mixed complex (Ω̄A+, b̃, B̃), where Ω̄
nA+ = A
⊗̂(n+1) ⊕A⊗̂n and
b 1− λ
0 −b′
; B̃ =
where λ(a1⊗· · ·⊗an) = (−1)
n−1(an⊗a1⊗· · ·⊗an−1) and N = id+λ+· · ·+λ
n−1 [16,
1.4.5]. The continuous Hochschild homology of A, the continuous cyclic homology
of A and the continuous periodic cyclic homology of A are defined by
HH∗(A) = H
∗(Ω̄A+, b̃, B̃), HC∗(A) = H
∗(Ω̄A+, b̃, B̃) and
HP∗(A) = H
∗ (Ω̄A+, b̃, B̃)
where Hb∗, H
∗ and H
∗ are Hochschild homology, cyclic homology and periodic cyclic
homology of the mixed complex (Ω̄A+, b̃, B̃) in LCS, see [17].
There is also a cyclic cohomology theory associated with a complete locally convex
algebra A, obtained when one replaces the chain complexes of A by their dual
complexes of strong dual spaces.
Lemma 4.1. (i) Let A be a [nuclear] Fréchet algebra. Then the following complexes
(C(A), b), (Ω̄A+, b̃), (BcΩ̄A+, b̃+ B̃) and (BpΩ̄A+, b̃+ B̃) are complexes of [nuclear]
Fréchet spaces and continuous linear operators.
(ii) Let A be a [nuclear] ⊗̂-algebra which is a DF -space. Then the following com-
plexes (C(A), b), (Ω̄A+, b̃), and (BcΩ̄A+, b̃+ B̃) are complexes of [nuclear] complete
DF -spaces and continuous linear operators, and (BpΩ̄A+, b̃ + B̃) is a complex of
[nuclear] complete locally convex spaces and continuous linear operators, but it is
not a DF -space in general.
Proof. It is well known that Fréchet spaces are closed under countable cartesian
products and projective tensor product [31]; nuclear locally convex spaces are closed
under cartesian products, countable direct sums and projective tensor product [12,
Corollary 21.2.3]; complete DF -spaces are closed under countable direct sums, pro-
jective tensor product, but not under infinite cartesian products [12, Theorem 12.4.8
and Theorem 15.6.2]. �
Propositions 4.2 and 4.3 below are proved by the author in [17, 18] and show
the equivalence between the continuous cyclic (co)homology of A and the contin-
uous periodic cyclic (co)homology of A when A has trivial continuous Hochschild
(co)homology HHn(A) for all n ≥ N for some integer N . Here we add in these
statements certain topological conditions on the algebra which allow us to show
that isomorphisms of (co)homology groups are automatically topological.
Proposition 4.2. [17, Proposition 3.2] Let A be a complete locally convex algebra.
Then, for any even integer N , say N = 2K, and the following assertions, we have
(i)N ⇒ (ii)N ⇒ (iii)N ⇒ (ii)N+1 and (ii)N ⇒ (iv)N :
10 Z. A. Lykova
(i)N H
naive
n (A) = {0} for all n ≥ N and H
n (A) = {0} for all n ≥ N − 1;
(ii)N HHn(A) = {0} for all n ≥ N ;
(iii)N for all k ≥ K, up to isomorphism of linear spaces,
HC2k(A) = HCN(A) and HC2k+1(A) = HCN−1(A);
(iv)N up to isomorphism of linear spaces, HP0(A) = HCN(A) and HP1(A) =
HCN−1(A).
For Fréchet algebras the isomorphisms in (iii)N and (iv)N are automatically topolog-
ical. For a nuclear ⊗̂-algebra A which is a DF -space the isomorphisms in (iii)N are
automatically topological.
Proof. A proof of the statement is given in [17, Proposition 3.2]. Here we add a
part on the automatic continuity of the isomorphisms. In view of the proofs of
[17, Propositions 2.1 and 3.2] it is easy to see that isomorphisms of homology in
(iii)N and (iv)N are induced by continuous morphisms of complexes. Note that by
Lemma 4.1, for a Fréchet algebra A, the following complexes (BcΩ̄A+, b̃ + B̃) and
(BpΩ̄A+, b̃ + B̃) are complexes of Fréchet spaces and continuous linear operators.
Thus, for Fréchet algebras, by [7, Lemma 0.5.9], isomorphisms of homology groups
are topological.
By Lemma 4.1, for a nuclear ⊗̂-algebra A which is a DF -space, the following com-
plex (BcΩ̄A+, b̃ + B̃) is a complex of nuclear complete DF -spaces and continuous
linear operators. By Corollary 3.6, for complete nuclear DF -spaces the isomor-
phisms for homology groups in (iii)N are also topological. �
Proposition 4.3. [18, Proposition 3.1] Let A be a complete locally convex algebra.
Then, for any even integer N , say N = 2K, and the following assertions, we have
(i)N ⇒ (ii)N ⇒ (iii)N ⇒ (ii)N+1 and (ii)N ⇒ (iv)N :
(i)N H
naive(A) = {0} for all n ≥ N and H
bar(A) = {0} for all n ≥ N − 1;
(ii)N for all n ≥ N, HH
n(A) = {0};
(iii)N for all k ≥ K, up to isomorphism of linear spaces, HC
2k(A) = HCN(A) and
HC2k+1(A) = HCN−1(A);
(iv)N up to isomorphism of linear spaces, HP
0(A) = HCN(A) and HP 1(A) =
HCN−1(A).
For nuclear Fréchet algebras the isomorphisms in (iii)N and (iv)N are topological
isomorphisms. For a nuclear ⊗̂-algebra A which is a DF -space the isomorphisms
in (iii)N are topological isomorphisms.
Proof. We need to add to the proof of [18, Proposition 3.1] the following part on
automatic continuity. In view of the proof of [18, Proposition 3.1] it is easy to
see that the isomorphisms of cohomology groups in (iii)N and (iv)N are induced by
continuous morphisms of complexes.
For nuclear Fréchet algebras, by Lemma 4.1, the complexes ((BcΩ̄A+)
∗, b̃∗ + B̃∗)
and ((BpΩ̄A+)
∗, b̃∗ + B̃∗) are complexes of strong duals of nuclear Fréchet spaces.
By Lemma 3.5, the isomorphisms of cohomology groups in (iii)N and (iv)N are
topological.
Cyclic cohomology of nuclear Fréchet and DF algebras 11
For a nuclear ⊗̂-algebra A which is a DF -space, by Lemma 4.1 and by [21,
Theorem 4.4.13], the chain complex (BcΩ̄A+, b̃+ B̃) is the strong dual of a complex
of nuclear Fréchet spaces. By [21, Theorem 4.4.12], complete nuclear DF -spaces and
nuclear Fréchet spaces are reflexive. Therefore ((BcΩ̄A+)
∗, b̃∗ + B̃∗) is a complex of
nuclear Fréchet spaces. Thus, by [7, Lemma 0.5.9], the isomorphisms of cohomology
groups in (iii)N are topological. �
The space of continuous traces on a topological algebra A is denoted by Atr, that
Atr = {f ∈ A∗ : f(ab) = f(ba) for all a, b ∈ A}.
The closure in A of the linear span of elements of the form {ab− ba : a, b ∈ A} is
denoted by [A,A]. Recall that b0 : A⊗̂A → A is uniquely determined by a ⊗ b 7→
ab− ba.
Proposition 4.4. Let A be in Φ and be a ⊗̂-algebra.
(i) Suppose that the continuous cohomology groups Hnaiven (A) = {0} for all n ≥ 1
and Hbarn (A) = {0} for all n ≥ 0. Then, up to isomorphism of linear spaces,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/Im b0;
HC2ℓ(A) = A/Im b0 and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
HP0(A) = A/Im b0 and HP1(A) = {0}.
(ii) Suppose that the continuous cohomology groups Hnnaive(A) = {0} for all n ≥ 1
and Hnbar(A) = {0} for all n ≥ 0. Then, up to isomorphism of linear spaces,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr;
HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
HP0(A) = Atr and HP1(A) = {0}.
Proof. (i). One can see that Hbarn (A) = {0} for all n ≥ 0 implies that
HHn(A) = H
naive
n (A) for all n ≥ 0,
see [17, Section 3]. Note that by definition of the ‘naive’ Hochschild homology of
A, Hnaive0 (A) = A/Im b0. Therefore, HHn(A) = {0} for all n ≥ 1 and HH0(A) =
A/Im b0.
From the exactness of the long Connes-Tsygan sequence of continuous homology
it follows that
HC0(A) = H
naive
0 (A) = A/Im b0 and HC1(A) = {0}.
The rest of Statement (i) follows from Proposition 4.2.
(ii) It is known that Hnbar(A) = {0} for all n ≥ 0, implies HH
n(A) = Hnnaive(A)
for all n ≥ 0. By definition of the ‘naive’ Hochschild cohomology of A, H0naive(A) =
Atr. Thus HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr.
From the exactness of the long Connes-Tsygan sequence of continuous cohomology
it follows that HC0(A) = H0naive(A) = A
tr and HC1(A) = {0}. The rest of
Statement (ii) follows from Proposition 4.3. �
12 Z. A. Lykova
5. Cyclic-type cohomology of biflat ⊗̂-algebras
Recall that a ⊗̂-algebra A is said to be biflat if it is flat in the category of A-⊗̂-
bimodules [7, Def. 7.2.5]. A ⊗̂-algebra A is said to be biprojective if it is projective
in the category of A-⊗̂-bimodules [7, Def. 4.5.1]. By [7, Proposition 4.5.6], a ⊗̂-
algebra A is biprojective if and only if there exists an A-⊗̂-bimodule morphism
ρA : A → A⊗̂A such that πA ◦ ρA = idA, where πA is the canonical morphism
πA : A⊗̂A → A, a1 ⊗ a2 7→ a1a2. It can be proved that any biprojective ⊗̂-algebra
is biflat and A = A2 = Im πA [7, Proposition 4.5.4]. Here A2 is the closure of
the linear span of the set {a1 · a2 : a1, a2 ∈ A} in A. A ⊗̂-algebra A is said to be
contractible if A+ is is projective in the category of A-⊗̂-bimodules. A ⊗̂-algebra A
is contractible if and only if A is biprojective and has an identity [7, Def. 4.5.8]. For
biflat Banach algebras A, Helemskii proved A = A2 = Im πA [7, Proposition 7.2.6]
and gave the description of the cyclic homology HC∗ and cohomology HC
∗ groups
of A in [8]. Later the author generalized Helemskii’s result to inverse limits of biflat
Banach algebras [17, Theorem 6.2] and to locally convex strict inductive limits of
amenable Banach algebras [18, Corollary 4.9].
Proposition 5.1. Let A be in Φ and be a biflat ⊗̂-algebra such that A = Im πA;
in particular, let A ∈ Φ be a biprojective ⊗̂-algebra. Then
(i) Hnaiven (A) = {0} for all n ≥ 1, H
naive
0 (A) = A/Im b0 and H
n (A) = {0} for all
n ≥ 0;
(ii) for the homology groups HH∗, HC∗ and HP∗ of A we have the isomorphisms of
linear spaces (5).
If, furthermore, A is a Fréchet space or A is a nuclear DF -space, then Hnaive0 (A) =
A/[A,A] is Hausdorff, and, for a biflat A, A = A2 implies that A = Im πA.
Proof. By [7, Theorem 3.4.25], up to topological isomorphism, the homology groups
Hnaiven (A) = Hn(A,A) = Tor
n (A,A+)
for all n ≥ 0. Since A is biflat, by [7, Proposition 7.1.2], Hnaiven (A) = {0} for all
n ≥ 1.
By [7, Theorem 3.4.26], up to topological isomorphism, the homology groups
Hbarn (A) = Hn+1(A,C) = Tor
n+1(C,C)
for all n ≥ 0, where C is the trivial A-bimodule. Note that, for the trivial A-
bimodule C, there is a flat resolution
0← C← A+ ← A← 0
in the category of left or right A-⊗̂-modules. By [7, Theorem 3.4.28], Hbarn (A) =
TorAn+1(C,C) = {0} for all n ≥ 1. By assumption, A = Im πA, hence H
0 (A) =
A/Im πA = {0}. Thus the conditions of Proposition 4.4 (i) are satisfied.
In the categories of Fréchet spaces and complete nuclear DF -spaces, the open
mapping theorem holds – see Corollary 3.4 for DF -spaces. Thus, by [7, Proposi-
tions 3.3.5 and 7.1.2], up to topological isomorphism, Hnaive0 (A) = Tor
0 (A,A+) is
Cyclic cohomology of nuclear Fréchet and DF algebras 13
Hausdorff. Since A is biflat, by [7, Proposition 7.1.2], TorA0 (C,A) is also Hausdorff.
By [7, Proposition 3.4.27], A2 = Im πA. �
A ⊗̂-algebra A is amenable if A+ is a flat A-⊗̂-bimodule. For a Fréchet algebra A
amenability is equivalent to the following: for all Fréchet A-bimodules X , H0(A, X)
is Hausdorff and Hn(A, X) = {0} for all n ≥ 1. Recall that an amenable Banach
algebra A is biflat and has a bounded approximate identity [7, Theorem VII.2.20].
Lemma 5.2. Let A be an amenable ⊗̂-algebra which is a Fréchet space or a nu-
clear DF -space. Then Hnaiven (A) = {0} for all n ≥ 1, H
naive
0 (A) = A/[A,A] and
Hbarn (A) = {0} for all n ≥ 0.
Proof. In the categories of Fréchet spaces and complete nuclear DF -spaces, the open
mapping theorem holds. Therefore, by [7, Theorem III.4.25 and Proposition 7.1.2],
up to topological isomorphism, for the trivial A-bimodule C,
Hbarn (A) = Hn+1(A,C) = Tor
n+1(C,A+) = {0}
for all n ≥ 0;
Hnaiven (A) = Hn(A,A) = Tor
n (A,A+) = {0}
for all n ≥ 1 and Hnaive0 (A) = Tor
0 (A,A+) is Hausdorff, that is, H
naive
0 (A) =
A/[A,A]. �
Theorem 5.3. Let A be a ⊗̂-algebra which is a Fréchet space or a nuclear DF -
space. Suppose that the continuous homology groups Hnaiven (A) = {0} for all n ≥ 1,
Hnaive0 (A) is Hausdorff and H
n (A) = {0} for all n ≥ 0. In particular, asssume
that A is a biflat algebra such that A = A2 or A is amenable. Then
(i) up to topological isomorphism,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/[A,A];
HC2ℓ(A) = A/[A,A] and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
(ii) up to topological isomorphism for Fréchet algebras and up to isomorphism of
linear spaces for nuclear DF -algebras,
(8) HP0(A) = A/[A,A] and HP1(A) = {0};
(iii)
Hnnaive(A) = {0} for all n ≥ 1;
Hnbar(A) = {0}; for all n ≥ 0;
(iv) up to topological isomorphism for nuclear Fréchet algebras and nuclear DF -
algebras and up to isomorphism of linear spaces for Fréchet algebras,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr;
HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
(v) up to topological isomorphism for nuclear Fréchet algebras and up to isomorphism
of linear spaces for Fréchet algebras and for nuclear DF -algebras,
(11) HP0(A) = Atr and HP1(A) = {0}.
14 Z. A. Lykova
Proof. In view of Proposition 5.1 and Lemma 5.2, a biflat algebra A such that
A = A2 and an amenable A satisfy the conditions of the theorem.
By Proposition 2.2, firstly, Hbarn (A) = {0} for all n ≥ 0 if and only if H
bar(A) =
{0} for all n ≥ 0; and, secondly, Hnnaive(A) = {0} for all n ≥ 1 if and only if
Hnaiven (A) = {0} for all n ≥ 1 and H
naive
0 (A) is Hausdorff.
By Proposition 4.4, we have isomorphisms of linear spaces in (i) – (v). In Propo-
sitions 4.2 and 4.2 we show also when the above isomorphisms are automatically
topological. �
Remark 5.4. Recall that, for a biflat Banach algebra A, dbwA ≤ 2 [27, Theorem
6]. By [14, Theorem 5.2], for a Banach algebra A of a finite weak bidimension dbwA,
we have isomorphisms between the entire cyclic cohomology and the periodic cyclic
cohomology of A, HE0(A) = HP 0(A) = Atr and HE1(A) = HP 1(A) = {0}. The
entire cyclic cohomology HEk(A) of A for k = 0, 1 are defined in [2, IV.7]. In
[25, Theorem 6.1] M. Puschnigg extended M. Khalkhali’s result on the isomorphism
HEk(A) = HP k(A) for k = 0, 1 from Banach algebras to some Fréchet algebras.
The following statement shows that the above theorems give the explicit descrip-
tion of cyclic type homology and cohomology of the projective tensor product of two
biprojective ⊗̂-algebras.
Proposition 5.5. Let B and C be biprojective ⊗̂-algebras. Then the projective
tensor product A = B⊗̂C is a biprojective ⊗̂-algebra.
Proof. Since B is biprojective, there is a morphism of B-⊗̂-bimodules ρB : B → B⊗̂B
such that πB ◦ ρB = idB. A similar statement is valid for C. Let i be the topological
isomorphism
i : (B⊗̂B)⊗̂(C⊗̂C)→ (B⊗̂C)⊗̂(B⊗̂C)
given by (b1⊗ b2)⊗ (c1⊗ c2) 7→ (b1⊗ c1)⊗ (b2⊗ c2). Note that πB⊗̂C = (πB⊗̂πC)◦ i
It is routine to check that
ρB⊗̂C : B⊗̂C → (B⊗̂C)⊗̂(B⊗̂C)
defined by ρB⊗̂C = i◦(ρB⊗ρC) is a morphism of B⊗̂C-⊗̂-bimodules and πB⊗̂C◦ρB⊗̂C =
idB⊗̂C. �
Remark 5.6. For amenable Banach algebras B and C, B. E. Johnson showed that
the Banach algebra A = B⊗̂C is amenable [11]. By [19, Proposition 5.4], for a biflat
Banach algebra A, each closed two-sided ideal I with bounded approximate identity
is amenable and the quotient algebra A/I is biflat. Thus the explicit description of
cyclic type homology and cohomology of such I and A/I is also given in Theorem
5.3. One can find a number of examples of biflat and simplicially trivial Banach and
C∗- algebras in [17, Example 4.6, 4.9].
Cyclic cohomology of nuclear Fréchet and DF algebras 15
6. Applications to the cyclic-type cohomology of biprojective
⊗̂-algebras
In this section we present examples of nuclear biprojective ⊗̂-algebras which are
Fréchet spaces or DF -spaces and the continuous cyclic-type homology and cohomol-
ogy of these algebras.
Example 6.1. Let G be a compact Lie group and let E(G) be the nuclear Fréchet
algebra of smooth functions on G with the convolution product. It was shown by
Yu.V. Selivanov that A = E(G) is biprojective [29].
Let E∗(G) be the strong dual to E(G), so that E∗(G) is a complete nuclear DF -
space. This is a ⊗̂-algebra with respect to convolution multiplication: for f, g ∈
E∗(G) and x ∈ E(G), < f ∗ g, x >=< f, y >, where y ∈ E(G) is defined by
y(s) =< g, xs >, s ∈ G and xs(t) = x(s
−1t), t ∈ G. J.L. Taylor proved that the
algebra of distributions E∗(G) on a compact Lie group G is contractible [30].
Example 6.2. Let (E, F ) be a pair of complete Hausdorff locally convex spaces
endowed with a jointly continuous bilinear form 〈·, ·〉 : E ×F → C that is not iden-
tically zero. The space A = E⊗̂F is a ⊗̂-algebra with respect to the multiplication
defined by
(x1 ⊗ x2)(y1 ⊗ y2) = 〈x2, y1〉x1 ⊗ y2, xi ∈ E, yi ∈ F.
Yu.V. Selivanov proved that this algebra is biprojective and usually non unital
[28, 29]. More exactly, if A = E⊗̂F has a left or right identity, then E or F
respectively is finite-dimensional. If the form 〈·, ·〉 is nondegenerate, then A = E⊗̂F
is called the tensor algebra generated by the duality (E, F, 〈·, ·〉).
In particular, if E is a Banach space with the approximation property, then the
algebra A = E⊗̂E∗ is isomorphic to the algebra N (E) of nuclear operators on E [7,
II.2.5].
6.1. Köthe sequence algebras. The following results on Köthe algebras can be
found in A. Yu. Pirkovskii’s papers [22, 23].
A set P of nonnegative real-valued sequences p = (pi)i∈N is called a Köthe set if
the following axioms are satisfied:
(P1) for every i ∈ N there is p ∈ P such that pi > 0;
(P2) for every p, q ∈ P there is r ∈ P such that max{pi, qi} ≤ ri for all i ∈ N.
Suppose, in addition, the following condition is satisfied:
(P3) for every p ∈ P there exist q ∈ P and a constant C > 0 such that pi ≤ Cq
for all i ∈ N.
For any Köthe set P which satisfies (P3), the Köthe space
λ(P ) = {x = (xn) ∈ C
N : ‖x‖p =
|xn|pn <∞ for all p ∈ P}
is a complete locally convex space with the topology determined by the family of
seminorms {‖x‖p : p ∈ P} and a ⊗̂-algebra with pointwise multiplication. The
⊗̂-algebras λ(P ) are called Köthe algebras.
By [21] and [6], for a Köthe set, λ(P ) is nuclear if and only if
(P4) for every p ∈ P there exist q ∈ P and ξ ∈ ℓ1 such that pi ≤ ξiqi for all i ∈ N.
16 Z. A. Lykova
By [22, Theorem 3.5], λ(P ) is biprojective if and only if
(P5) for every p ∈ P there exist q ∈ P and a constant M > 0 such that p2i ≤Mqi
for all i ∈ N.
The algebra λ(P ) is unital if and only if
n pn <∞ for every p ∈ P .
Example 6.3. Fix a real number 1 ≤ R ≤ ∞ and a nondecreasing sequence
α = (αi) of positive numbers with limi→∞ αi =∞. The power series space
ΛR(α) = {x = (xn) ∈ C
N : ‖x‖r =
|xn|r
αn <∞ for all 0 < r < R}
is a Fréchet Köthe algebra with pointwise multiplication. The topology of ΛR(α) is
determined by a countable family of seminorms {‖x‖rk : k ∈ N} where {rk} is an
arbitrary increasing sequence converging to R.
By [23, Corollary 3.3], ΛR(α) is biprojective if and only if R = 1 or R =∞.
By the Grothendieck-Pietsch criterion, ΛR(α) is nuclear if and only if for limn
0 for R <∞ and limn
<∞ for R =∞, see [22, Example 3.4].
The algebra ΛR((n)) is topologically isomorphic to the algebra of functions holo-
morphic on the open disc of radius R, endowed with Hadamard product, that is,
with “co-ordinatewise” product of the Taylor expansions of holomorphic functions.
Example 6.4. The algebra H(C) ∼= Λ∞((n)) of entire functions, endowed with the
Hadamard product, is a biprojective nuclear Fréchet algebra [23].
Example 6.5. The algebra H(D1) ∼= Λ1((n)) of functions holomorphic on the open
unit disc, endowed with the Hadamard product, is a biprojective nuclear Fréchet
algebra. Moreover it is contractible, since the function z 7→ (1− z)−1 is an identity
for H(D1) [23].
For any Köthe space λ(P ) the dual space λ(P )∗ can be canonically identified with
{(yn) ∈ C
N : ∃p ∈ P and C > 0 such that |yn| ≤ Cpn for all n ∈ N}.
It is shown in [23] that, for a biprojective Köthe algebra λ(P ), λ(P )∗ is a sequence
algebra with pointwise multiplication.
The algebra λ(P )∗ is unital if and only if there exists p ∈ P such that inf i pi > 0.
Example 6.6. The nuclear Fréchet algebra of rapidly decreasing sequences
s = {x = (xn) ∈ C
N : ‖x‖k =
|xn|n
k <∞ for all k ∈ N}
is a biprojective Köthe algebra [22]. The algebra s is topologically isomorphic to
Λ∞(α) with αn = logn [23]. The nuclear Köthe ⊗̂-algebra s
∗ of sequences of poly-
nomial growth is contractible [30].
Example 6.7. [23, Section 4.2] Let P be a Köthe set such that pi ≥ 1 for all p ∈ P
and all n ∈ N. Then the formula 〈a, b〉 =
i aibi defines a jointly continuous,
nondegenerate bilinear form on λ(P ) × λ(P ). Thus M(P ) = λ(P )⊗̂λ(P ) can be
considered as the tensor algebra generated by the duality (λ(P ), λ(P ), 〈·, ·〉), and so
is biprojective. There is a canonical isomorphism between M(P ) and the algebra
Cyclic cohomology of nuclear Fréchet and DF algebras 17
λ(P × P ) of N×N complex matrices (aij)(ij)∈N×N satisfying the condition ‖a‖p =
i,j |aij|pipj <∞ for all p ∈ P with the usual matrix multiplication.
In particular, for P = {(nk)n∈N : k = 0, 1, . . . }, we obtain the biprojective nuclear
Fréchet algebra ℜ = s⊗̂s of “smooth compact operators” consisting of N×N com-
plex matrices (aij) with rapidly decreasing matrix entries. Here s is from Example
Theorem 6.8. Let A be a ⊗̂-algebra belonging to one of the following classes:
(i) A = E(G) or A = E∗(G) for a compact Lie group G;
(ii) A = E⊗̂F , the tensor algebra generated by the duality (E, F, 〈·, ·〉) for nuclear
Fréchet spaces E and F (e.g., ℜ = s⊗̂s) or for nuclear complete DF -spaces E and
(iii) Fréchet Köthe algebras A = λ(P ) such that the Köthe set P satisfies (P3),
(P4) and (P5); in particular, Λ1(α) such that limn
= 0 or Λ∞(α) such that
<∞. (e.g., H(D1), s, H(C)).
(iv) Köthe algebras A = λ(P )∗ which are the strong duals of λ(P ) from (iii).
(v) the projective tensor product A = B⊗̂C of biprojective nuclear ⊗̂-algebras B
and C which are Fréchet spaces or DF -spaces; in particular, A = E(G)⊗̂ℜ.
Then, up to topological isomorphism,
Hnaiven (A) = {0} for all n ≥ 1 and H
naive
0 (A) = A/[A,A];
Hbarn (A) = {0} for all n ≥ 0;
HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/[A,A];
HC2ℓ(A) = A/[A,A] and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
Hnnaive(A) = {0} for all n ≥ 1; H
bar(A) = {0} for all n ≥ 0;
HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr;
HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
and, up to topological isomorphism for Fréchet algebras and up to isomorphism of
linear spaces for DF -algebras,
HP0(A) = A/[A,A] and HP1(A) = {0};
HP0(A) = Atr and HP1(A) = {0}.
Proof. We have mentioned above that the algebras in (i)-(iii) and (v) are biprojective
and nuclear. By [23, Corollary 3.10], for any nuclear biprojective Fréchet Köthe
algebra λ(P ), the strong dual λ(P )∗ is a nuclear, biprojective Köthe ⊗̂-algebra
which is a DF -space. For nuclear Fréchet algebras and for nuclear DF -algebras, the
conditions of Theorem 5.3 are satisfied. Therefore, for the homology and cohomology
groups HH and HC of A we have the topological isomorphisms (7) and (10). For
the periodic cyclic homology and cohomology groups HP of A, for Fréchet algebras,
we have topological isomorphisms and, for nuclear DF -algebras, isomorphisms of
linear spaces (8) and (11). It is obvious that, for commutative algebras, Atr = A∗
and A/[A,A] = A. �
18 Z. A. Lykova
The cyclic-type homology and cohomology of E(G) for a compact Lie group G
were calculated in [20].
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[15] G. Köthe: Topological Vector Spaces, I, Die Grundlehren der mathematischen Wissenschaften,
159, Springer Verlag, Berlin, New-York, 1979.
[16] J.-L. Loday: Cyclic Homology, Springer Verlag, Berlin, 1992.
[17] Z.A. Lykova: “Cyclic cohomology of projective limits of topological algebras”, Proc. Edin-
burgh Math. Soc., Vol. 49, (2006), pp. 173–199.
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Appl. Algebra, Vol. 205, (2006), pp. 471–497.
[19] Z.A. Lykova and M.C. White: “Excision in the cohomology of Banach algebras with coeffi-
cients in dual bimodules”, In: E. Albrecht and M. Mathieu (Eds.), Banach Algebras’97, Walter
de Gruyter Publishers, Berlin, 1998, pp. 341–361.
[20] R. Meyer: “Comparisons between periodic, analytic and local cyclic cohomology”,
ArXiv:math.KT/0205276 v2 18 Dec 2003.
[21] A. Pietsch: Nuclear Locally Convex Spaces, Springer Verlag, Berlin, 1972.
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Sci., Vol. 111, No. 2, (2001), pp. 3476-3495.
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301–364. (Russian with English summary)
http://arxiv.org/abs/math/0205276
Cyclic cohomology of nuclear Fréchet and DF algebras 19
[25] M. Puschnigg: “Excision in cyclic homology theories”, Invent. Math., Vol. 143, (2001), pp.
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, Vol. 307, (1988), pp. 329–334.
School of Mathematics and Statistics, University of Newcastle,, Newcastle
upon Tyne, NE1 7RU, UK (Z.A.Lykova@newcastle.ac.uk)
	1. Introduction
	2. Definitions and notation
	3. The open mapping theorem in complete nuclear DF-spaces
	4. Cyclic and Hochschild cohomology of some -algebras
	5. Cyclic-type cohomology of biflat -algebras
	6. Applications to the cyclic-type cohomology of biprojective -algebras
	6.1.  Köthe sequence algebras
	References

  We give explicit formulae for the continuous Hochschild and cyclic homology
and cohomology of certain topological algebras. To this end we show that, for a
continuous morphism $\phi: \X\to \Y$ of complexes of complete nuclear
$DF$-spaces, the isomorphism of cohomology groups $H^n(\phi): H^n(\X) \to
H^n(\Y)$ is automatically topological. The continuous cyclic-type homology and
cohomology are described up to topological isomorphism for the following
classes of biprojective $\hat{\otimes}$-algebras: the tensor algebra $E
\hat{\otimes} F$ generated by the duality $(E, F, < \cdot, \cdot >)$ for
nuclear Fr\'echet spaces $E$ and $F$ or for nuclear $DF$-spaces $E$ and $F$;
nuclear biprojective K\"{o}the algebras $\lambda(P)$ which are Fr\'echet spaces
or $DF$-spaces; the algebra of distributions $\mathcal{E}^*(G)$ on a compact
Lie group $G$.

Introduction
Cyclic cohomology groups of topological algebras play an essential role in non-
commutative geometry [2]. There has been a number of papers addressing the
calculation of cyclic-type continuous homology and cohomology groups of some Ba-
nach, C∗- and topological algebras; see, e.g., [2, 8, 14, 17, 18, 20, 32]. However, it
remains difficult to describe these groups explicitly for many topological algebras.
To compute the continuous Hochschild and cyclic cohomology groups of Fréchet
algebras one has to deal with complexes of complete DF -spaces. Here, in addition
to presenting known homological techniques we also supply technical enhancements
that permit the necessary generalization of results known in the case of Banach
algebras and their inverse limits to wider classes of topological algebras notably to
those that occur in noncommutative geometry.
The category of Banach spaces has the useful property that it is closed under
passage to dual spaces. Fréchet spaces do not have this property: the strong dual
Date: 22 August 2007.
Key words and phrases. Cyclic cohomology, Hochschild cohomology, nuclear DF -spaces, locally
convex algebras, nuclear Fréchet algebra.
I am indebted to the Isaac Newton Institute for Mathematical Sciences at Cambridge for hos-
pitality and for generous financial support from the programme on Noncommutative Geometry
while this work was carried out.
http://arxiv.org/abs/0704.1019v2
2 Z. A. Lykova
of a Fréchet space is a complete DF -space. DF -spaces have the awkward feature
that their closed subspaces need not be DF -spaces. However, closed subspaces of
complete nuclear DF -spaces are again DF -spaces [21, Proposition 5.1.7].
In Section 3 we use the strongest known results on the open mapping theorem to
give sufficient conditions on topological spaces E and F to imply that any continuous
linear operator T from E∗ onto F ∗ is open. This allows us to prove the following
results. In Lemma 3.6 we show that, for a continuous morphism ϕ : X → Y of
complexes of complete nuclear DF -spaces, the isomorphism of cohomology groups
Hn(ϕ) : Hn(X )→ Hn(Y) is automatically topological.
We use this fact to describe explicitly up to topological isomorphism the continuous
Hochschild and cyclic cohomology groups of nuclear ⊗̂-algebras A which are Fréchet
spaces or DF -spaces and have trivial Hochschild homology HHn(A) for all n ≥ 1
(Theorem 5.3). In Proposition 4.4, under the same condition on HHn(A), we give
explicit formulae, up to isomorphism of linear spaces, for continuous cyclic-type
homology of A in a more general category of underlying spaces.
In Theorem 6.8 the continuous cyclic-type homology and cohomology groups are
described up to topological isomorphism for the following classes of biprojective ⊗̂-
algebras: the tensor algebra E⊗̂F generated by the duality (E, F, 〈·, ·〉) for nuclear
Fréchet spaces or for nuclear complete DF -spaces E and F ; nuclear biprojective
Fréchet Köthe algebras λ(P ); nuclear biprojective Köthe algebras λ(P )∗ which are
DF -spaces; the algebra of distributions E∗(G) and the algebra of smooth functions
E(G) on a compact Lie group G.
2. Definitions and notation
We recall some notation and terminology used in homology and in the theory of
topological algebras. Homological theory can be found in any relevant textbook, for
instance, Loday [16] for the pure algebraic case and Helemskii [7] for the continuous
case.
Throughout the paper ⊗̂ is the projective tensor product of complete locally
convex spaces, by X⊗̂n we mean the n-fold projective tensor power X⊗̂ . . . ⊗̂X of
X , and id denotes the identity operator.
We use the notation Ban, Fr and LCS for the categories whose objects are Banach
spaces, Fréchet spaces and complete Hausdorff locally convex spaces respectively,
and whose morphisms in all cases are continuous linear operators. For topological
homology theory it is important to find a suitable category for the underlying spaces
of the algebras and modules. In [7] Helemskii constructed homology theory for the
following categories Φ of underlying spaces, for which he used the notation (Φ, ⊗̂).
Definition 2.1. ([7, Section II.5]) A suitable category for underlying spaces of
the algebras and modules is an arbitrary complete subcategory Φ of LCS having the
following properties:
(i) if Φ contains a space, it also contains all those spaces topologically isomorphic
to it;
Cyclic cohomology of nuclear Fréchet and DF algebras 3
(ii) if Φ contains a space, it also contains any of its closed subspaces and the
completion of any its Hausdorff quotient spaces;
(iii) Φ contains the direct sum and the projective tensor product of any pair of its
spaces;
(iv) Φ contains C.
Besides Ban, Fr and LCS important examples of suitable categories Φ are the
categories of complete nuclear spaces [31, Proposition 50.1], nuclear Fréchet spaces
and complete nuclear DF -spaces. As to the above properties for the category of
complete nuclear DF -spaces, recall the following results. By [12, Theorem 15.6.2],
if E and F are complete DF -spaces, then E⊗̂F is a complete DF -space. By [21,
Proposition 5.1.7], a closed linear subspace of a complete nuclear DF -space is also
a complete nuclear DF -space. By [21, Proposition 5.1.8], each quotient space of a
complete nuclear DF -space by a closed linear subspace is also a complete nuclear
DF -space.
By definition a ⊗̂-algebra is a complete Hausdorff locally convex algebra with
jointly continuous multiplication. A left ⊗̂-module X over a ⊗̂-algebra A is a com-
plete Hausdorff locally convex spaceX together with the structure of a leftA-module
such that the map A×X → X , (a, x) 7→ a ·x is jointly continuous. For a ⊗̂-algebra
A, ⊗̂A is the projective tensor product of left and right A-⊗̂-modules (see [6], [7,
II.4.1]). The category of left A-⊗̂-modules is denoted by A-mod and the category
of A-⊗̂-bimodules is denoted by A-mod-A.
Let K be one of the above categories. A chain complex X∼ in the category K is a
sequence of Xn ∈ K and morphisms dn
· · · ← Xn
← Xn+1
← Xn+2 ← . . .
such that dn ◦ dn+1 = 0 for every n. The homology groups of X∼ are defined by
Hn(X∼) = Ker dn−1/Im dn.
A continuous morphism of chain complexes ψ∼ : X∼ → P∼ induces a continuous
linear operator Hn(ψ∼) : Hn(X∼)→ Hn(P∼) [9, Definition 0.4.22].
If E is a topological vector space E∗ denotes its dual space of continuous linear
functionals. Throughout the paper, E∗ will always be equipped with the strong
topology unless otherwise stated. The strong topology is defined on E∗ by taking as
a basis of neighbourhoods of 0 the family of polars V 0 of all bounded subsets V of
E; see [31, II.19.2].
For any ⊗̂-algebra A, not necessarily unital, A+ is the ⊗̂-algebra obtained by
adjoining an identity to A. For a ⊗̂-algebra A, the algebra Ae = A+⊗̂A
called the enveloping algebra of A, where A
+ is the opposite algebra of A+ with
multiplication a · b = ba.
A complex of A-⊗̂-modules and their morphisms is called admissible if it splits
as a complex in LCS [7, III.1.11]. A module Y ∈ A-mod is called flat if for any
admissible complex X of right A-⊗̂-modules the complex X⊗̂AY is exact. A module
Y ∈ A-mod-A is called flat if for any admissible complex X of A-⊗̂-bimodules the
4 Z. A. Lykova
complex X⊗̂AeY is exact. For Y,X ∈ A-mod-A, we shall denote by Tor
n (X, Y ) the
nth homology of the complex X⊗̂AeP, where 0← Y ← P is a projective resolution
of Y in A-mod-A, [7, Definition III.4.23].
It is well known that the strong dual of a Fréchet space is a complete DF -space
and that nuclear Fréchet spaces and complete nuclear DF -spaces are reflexive [21,
Theorem 4.4.12]. Moreover, the correspondence E ↔ E∗ establishes a one-to-one
relation between nuclear Fréchet spaces and complete nuclear DF -spaces [21, The-
orem 4.4.13]. DF -spaces were introduced by A. Grothendieck in [5].
Further we shall need the following technical result which extends a result of
Johnson for the Banach case [11, Corollary 1.3].
Proposition 2.2. Let (X , d) be a chain complex of
(a) Fréchet spaces and continuous linear operators, or
(b) complete nuclear DF -spaces and continuous linear operators,
and let N ∈ N. Then the following statements are equivalent:
(i) Hn(X , d) = {0} for all n ≥ N and HN−1(X , d) is Hausdorff;
(ii) Hn(X ∗, d∗) = {0} for all n ≥ N.
Proof. Recall that Hn(X , d) = Ker dn−1/Im dn and H
n(X ∗, d∗) = Ker d∗n/Im d
Let L be the closure of Im dN−1 in XN−1. Consider the following commutative
diagram
0 ← L
←− XN
←− XN+1
←− . . .
↓ i ւ dN−1
in which i is the natural inclusion and j is a corestriction of dN−1. The dual com-
mutative diagram is the following
0 → L∗
−→ X∗N
−→ X∗N+1
−→ . . .
↑ i∗ ր d∗N−1
X∗N−1
It is clear that HN−1(X , d) is Hausdorff if and only if j is surjective. Since i is
injective, condition (i) is equivalent to the exactness of diagram (1). On the other
hand, by the Hahn-Banach theorem, i∗ is surjective. Thus condition (ii) is equivalent
to the exactness of diagram (2).
In the case of Fréchet spaces, by [18, Lemma 2.3], the exactness of the complex
(1) is equivalent to the exactness of the complex (2).
In the case of complete nuclear DF -spaces, by [21, Proposition 5.1.7], L is the
strong dual of a nuclear Fréchet space. By [21, Theorem 4.4.12], complete nuclear
DF -spaces are reflexive, and therefore the complex (1) is the dual of the complex
(2) of nuclear Fréchet spaces and continuous linear operators. By [18, Lemma 2.3],
the exactness of the complex (1) is equivalent to the exactness of the complex (2).
The proposition is proved. �
Cyclic cohomology of nuclear Fréchet and DF algebras 5
3. The open mapping theorem in complete nuclear DF -spaces
It is known that there exist closed linear subspaces of DF -spaces that are not
DF -spaces. For nuclear spaces, however, we have the following.
Lemma 3.1. [21, Proposition 5.1.7] Each closed linear subspace F of the strong dual
E∗ of a nuclear Fréchet space E is also the strong dual of a nuclear Fréchet space.
In a locally convex space a subset is called a barrel if it is absolutely convex,
absorbent and closed. Every locally convex space has a neighbourhood base con-
sisting of barrels. A locally convex space is called a barrelled space if every barrel
is a neighbourhood [26]. By [26, Theorem IV.1.2], every Fréchet space is barrelled.
By [26, Corollary IV.3.1], a Hausdorff locally convex space is reflexive if and only
if it is barrelled and every bounded set is contained in a weakly compact set. Thus
the strong dual of a nuclear Fréchet space is barrelled. For a generalization of the
open mapping theorem to locally convex spaces, V. Pták introduced the notion of
B-completeness in [24]. A subspace Q of E∗ is said to be almost closed if, for each
neighbourhood U of 0 in E, Q∩U0 is closed in the relative weak* topology σ(E∗, E)
on U0. A locally convex space E is said to be B-complete or fully complete if each
almost closed subspace of E∗ is closed in the weak* topology σ(E∗, E).
Theorem 3.2. [24]. Let E be a B-complete locally convex space and F be a barrelled
locally convex space. Then a continuous linear operator f of E onto F is open.
Recall [10, Theorem 4.1.1] that a locally convex space E is B-complete if and only
if each linear continuous and almost open mapping f of E onto any locally convex
space F is open. By [10, Proposition 4.1.3], every Fréchet space is B-complete.
Theorem 3.3. Let E be a semi-reflexive metrizable barrelled space, F be a Hausdorff
reflexive locally convex space and let E∗ and F ∗ be the strong duals of E and F
respectively. Then a continuous linear operator T of E∗ onto F ∗ is open.
Proof. By [10, Theorem 6.5.10] and by [10, Corollary 6.2.1], the strong dual E∗
of a semi-reflexive metrizable barrelled space E is B-complete. By [26, Corollary
IV.3.2], if a Hausdorff locally convex space is reflexive, so is its dual under the strong
topology. By [26, Corollary IV.3.1], a Hausdorff reflexive locally convex space is
barrelled. Hence F ∗ is a barrelled locally convex space. Therefore, by Theorem 3.2,
T is open. �
Corollary 3.4. Let E and F be nuclear Fréchet spaces and let E∗ and F ∗ be the
strong duals of E and F respectively. Then a continuous linear operator T of E∗
onto F ∗ is open.
For a continuous morphism of chain complexes ψ∼ : X∼ → P∼ in Fr, a surjective
map Hn(ϕ) : Hn(X ) → Hn(Y) is automatically open, see [7, Lemma 0.5.9]. To get
the corresponding result for dual complexes of Fréchet spaces one has to assume
nuclearity.
6 Z. A. Lykova
Lemma 3.5. Let (X , dX ) and (Y , dY) be chain complexes of nuclear Fréchet spaces
and continuous linear operators and let (X ∗, d∗X ) and (Y
∗, d∗Y) be their strong dual
complexes. Let ϕ : X ∗ → Y∗ be a continuous morphism of complexes. Suppose that
ϕ∗ = H
n(ϕ) : Hn(X ∗, d∗X )→ H
n(Y∗, d∗Y)
is surjective. Then ϕ∗ is open.
Proof. Let σY∗ : Ker (d
Y)n → H
n(Y∗, d∗Y) be the quotient map. Consider the map
ψ : Ker (d∗X )n ⊕ Y
n−1 → Ker (d
Y)n ⊂ Y
given by (x, y) 7→ ϕn(x) + (d
Y)n−1(y).
By Lemma 3.1, Ker (d∗X )n and Ker (d
Y)n are the strong duals of nuclear Fréchet
spaces and hence are barrelled. By [10, Theorem 6.5.10] and [10, Corollary 6.2.1],
the strong dual of a semi-reflexive metrizable barrelled space is B-complete. Thus
Ker (d∗X )n, Y
n−1 and
Ker (d∗X )n ⊕ Y
∼= [(Ker (d∗X )n)
∗ ⊕ Yn−1]
are B-complete. By assumption ϕ∗ maps H
n(X ∗, d∗X ) onto H
n(Y∗, d∗Y), which im-
plies that ψ is a surjective linear continuous operator from the B-complete locally
convex space Ker (d∗X )n ⊕ Y
n−1 to the barrelled locally convex space Ker (d
Therefore, by Theorem 3.2, ψ is open. Consider the diagram
Ker (d∗X )n ⊕ Y
→ Ker (d∗X )n
→ Hn(X ∗, d∗X )
↓ ψ ↓ ϕ∗
Ker (d∗Y)n
−→ Hn(Y∗, d∗Y)
in which j is a projection onto a direct summand and σX ∗ and σY∗ are the natural
quotient maps. Obviously this diagram is commutative. Note that the projection j
and quotient maps σX ∗ , σY∗ are open. As ψ is also an open map, so is σY∗ ◦ ψ =
ϕ∗ ◦ σX ∗ ◦ j. Since σX ∗ ◦ j is continuous, ϕ∗ is open. �
Corollary 3.6. Let (X , dX ) and (Y , dY) be cochain complexes of complete nuclear
DF -spaces and continuous linear operators, and let ϕ : X → Y be a continuous
morphism of complexes. Suppose that ϕ∗ = H
n(ϕ) : Hn(X , dX ) → H
n(Y , dY) is
surjective. Then ϕ∗ is open.
Proof. By [21, Theorem 4.4.13], (X , dX ) and (Y , dY) are strong duals of chain com-
plexes (X ∗, d∗X ) and (Y
∗, d∗Y) of nuclear Fréchet spaces and continuous operators.
The result follows from Lemma 3.5. �
4. Cyclic and Hochschild cohomology of some ⊗̂-algebras
One can consult the books by Loday [16] or Connes [2] on cyclic-type homological
theory.
Let A be a ⊗̂-algebra and let X be an A-⊗̂-bimodule. We assume here that the
category of underlying spaces Φ has the properties from Definition 2.1. Let us recall
the definition of the standard homological chain complex C∼(A, X). For n ≥ 0, let
Cyclic cohomology of nuclear Fréchet and DF algebras 7
Cn(A, X) denote the projective tensor product X⊗̂A
. The elements of Cn(A, X)
are called n-chains. Let the differential dn : Cn+1 → Cn be given by
dn(x⊗ a1 ⊗ . . .⊗ an+1) = x · a1 ⊗ . . .⊗ an+1+
(−1)k(x⊗ a1 ⊗ . . .⊗ akak+1 ⊗ . . .⊗ an+1) + (−1)
n+1(an+1 · x⊗ a1 ⊗ . . .⊗ an)
with d−1 the null map. The homology groups of this complex Hn(C∼(A, X)) are
called the continuous Hochschild homology groups of A with coefficients in X and
denoted by Hn(A, X) [7, Definition II.5.28]. We also consider the cohomology
groups Hn((C∼(A, X))
∗) of the dual complex (C∼(A, X))
∗ with the strong dual
topology. For Banach algebras A, Hn((C∼(A, X))
∗) is topologically isomorphic to
the Hochschild cohomologyHn(A, X∗) of A with coefficients in the dual A-bimodule
X∗ [7, Definition I.3.2 and Proposition II.5.27]. The weak bidimension of a Fréchet
algebra A is
dbwA = inf{n : H
n+1(C∼(A, X)
∗) = {0} for all Fréchet A−bimodules X}.
The continuous bar and ‘naive’ Hochschild homology of a ⊗̂-algebra A are defined
respectively as
Hbar∗ (A) = H∗(C(A), b
′) and Hnaive∗ (A) = H∗(C(A), b),
where Cn(A) = A
⊗̂(n+1), and the differentials b, b′ are given by
b′(a0 ⊗ · · · ⊗ an) =
(−1)i(a0 ⊗ · · · ⊗ aiai+1 ⊗ · · · ⊗ an) and
b(a0 ⊗ · · · ⊗ an) = b
′(a0 ⊗ · · · ⊗ an) + (−1)
n(ana0 ⊗ · · · ⊗ an−1).
Note that Hnaive∗ (A) is just another way of writing H∗(A,A), the continuous homol-
ogy of A with coefficients in A, as described in [7, 11].
There is a powerful method based on mixed complexes for the study of the cyclic-
type homology groups; see papers by C. Kassel [13], J. Cuntz and D. Quillen [4] and
J. Cuntz [3]. We shall present this method for the category LCS of locally convex
spaces and continuous linear operators; see [1] for the category of Fréchet spaces. A
mixed complex (M, b, B) in the category LCS is a familyM = {Mn}n≥0 of locally
convex spaces Mn equipped with continuous linear operators bn : Mn → Mn−1 and
Bn : Mn → Mn+1, which satisfy the identities b
2 = bB + Bb = B2 = 0. We assume
that in degree zero the differential b is identically equal to zero. We arrange the
8 Z. A. Lykova
mixed complex (M, b, B) in the double complex
. . . . . . . . . . . .
b ↓ b ↓ b ↓
b ↓ b ↓
There are three types of homology theory that can be naturally associated with a
mixed complex. The Hochschild homology Hb∗(M) of (M, b, B) is the homology of
the chain complex (M, b), that is,
Hbn(M) = Hn(M, b) = Ker {bn :Mn →Mn−1}/Im {bn+1 :Mn+1 →Mn}.
To define the cyclic homology of (M, b, B), let us denote by BcM the total complex
of the above double complex, that is,
· · · → (BcM)n
→ (BcM)n−1 → · · ·
→ (BcM)0 → 0,
where the spaces
(BcM)0 =M0, . . . , (BcM)2k−1 =M1 ⊕M3 ⊕ · · · ⊕M2k−1
(BcM)2k =M0 ⊕M2 ⊕ · · · ⊕M2k
are equipped with the product topology, and the continuous linear operators b+B
are defined by
(b+B)(y0, . . . , y2k) = (by2 +By0, . . . , by2k +By2k−2)
(b+B)(y1, . . . , y2k+1) = (by1, . . . , by2k+1 +By2k−1).
The cyclic homology of (M, b, B) is defined to be H∗(BcM, b+B). It is denoted by
Hc∗(M, b, B).
The periodic cyclic homology of (M, b, B) is defined in terms of the complex
· · · → (BpM)ev
→ (BpM)odd
→ (BpM)ev
→ (BpM)odd → . . . ,
where even/odd chains are elements of the product spaces
(BpM)ev =
M2n and (BpM)odd =
M2n+1,
respectively. The spaces (BpM)ev/odd are locally convex spaces with respect to the
product topology [15, Section 18.3.(5)]. The continuous differential b+B is defined
as an obvious extension of the above. The periodic cyclic homology of (M, b, B) is
Hpν (M, b, B) = Hν(BpM, b+B), where ν ∈ Z/2Z.
There are also three types of cyclic cohomology theory associated with the mixed
complex, obtained when one replaces the chain complex of locally convex spaces
Cyclic cohomology of nuclear Fréchet and DF algebras 9
by its dual complex of strong dual spaces. For example, the cyclic cohomology
associated with the mixed complex (M, b, B) is defined to be the cohomology of
the dual complex ((BcM)
∗, b∗ + B∗) of strong dual spaces and dual operators; it is
denoted by H∗c (M
∗, b∗, B∗).
Consider the mixed complex (Ω̄A+, b̃, B̃), where Ω̄
nA+ = A
⊗̂(n+1) ⊕A⊗̂n and
b 1− λ
0 −b′
; B̃ =
where λ(a1⊗· · ·⊗an) = (−1)
n−1(an⊗a1⊗· · ·⊗an−1) and N = id+λ+· · ·+λ
n−1 [16,
1.4.5]. The continuous Hochschild homology of A, the continuous cyclic homology
of A and the continuous periodic cyclic homology of A are defined by
HH∗(A) = H
∗(Ω̄A+, b̃, B̃), HC∗(A) = H
∗(Ω̄A+, b̃, B̃) and
HP∗(A) = H
∗ (Ω̄A+, b̃, B̃)
where Hb∗, H
∗ and H
∗ are Hochschild homology, cyclic homology and periodic cyclic
homology of the mixed complex (Ω̄A+, b̃, B̃) in LCS, see [17].
There is also a cyclic cohomology theory associated with a complete locally convex
algebra A, obtained when one replaces the chain complexes of A by their dual
complexes of strong dual spaces.
Lemma 4.1. (i) Let A be a [nuclear] Fréchet algebra. Then the following complexes
(C(A), b), (Ω̄A+, b̃), (BcΩ̄A+, b̃+ B̃) and (BpΩ̄A+, b̃+ B̃) are complexes of [nuclear]
Fréchet spaces and continuous linear operators.
(ii) Let A be a [nuclear] ⊗̂-algebra which is a DF -space. Then the following com-
plexes (C(A), b), (Ω̄A+, b̃), and (BcΩ̄A+, b̃+ B̃) are complexes of [nuclear] complete
DF -spaces and continuous linear operators, and (BpΩ̄A+, b̃ + B̃) is a complex of
[nuclear] complete locally convex spaces and continuous linear operators, but it is
not a DF -space in general.
Proof. It is well known that Fréchet spaces are closed under countable cartesian
products and projective tensor product [31]; nuclear locally convex spaces are closed
under cartesian products, countable direct sums and projective tensor product [12,
Corollary 21.2.3]; complete DF -spaces are closed under countable direct sums, pro-
jective tensor product, but not under infinite cartesian products [12, Theorem 12.4.8
and Theorem 15.6.2]. �
Propositions 4.2 and 4.3 below are proved by the author in [17, 18] and show
the equivalence between the continuous cyclic (co)homology of A and the contin-
uous periodic cyclic (co)homology of A when A has trivial continuous Hochschild
(co)homology HHn(A) for all n ≥ N for some integer N . Here we add in these
statements certain topological conditions on the algebra which allow us to show
that isomorphisms of (co)homology groups are automatically topological.
Proposition 4.2. [17, Proposition 3.2] Let A be a complete locally convex algebra.
Then, for any even integer N , say N = 2K, and the following assertions, we have
(i)N ⇒ (ii)N ⇒ (iii)N ⇒ (ii)N+1 and (ii)N ⇒ (iv)N :
10 Z. A. Lykova
(i)N H
naive
n (A) = {0} for all n ≥ N and H
n (A) = {0} for all n ≥ N − 1;
(ii)N HHn(A) = {0} for all n ≥ N ;
(iii)N for all k ≥ K, up to isomorphism of linear spaces,
HC2k(A) = HCN(A) and HC2k+1(A) = HCN−1(A);
(iv)N up to isomorphism of linear spaces, HP0(A) = HCN(A) and HP1(A) =
HCN−1(A).
For Fréchet algebras the isomorphisms in (iii)N and (iv)N are automatically topolog-
ical. For a nuclear ⊗̂-algebra A which is a DF -space the isomorphisms in (iii)N are
automatically topological.
Proof. A proof of the statement is given in [17, Proposition 3.2]. Here we add a
part on the automatic continuity of the isomorphisms. In view of the proofs of
[17, Propositions 2.1 and 3.2] it is easy to see that isomorphisms of homology in
(iii)N and (iv)N are induced by continuous morphisms of complexes. Note that by
Lemma 4.1, for a Fréchet algebra A, the following complexes (BcΩ̄A+, b̃ + B̃) and
(BpΩ̄A+, b̃ + B̃) are complexes of Fréchet spaces and continuous linear operators.
Thus, for Fréchet algebras, by [7, Lemma 0.5.9], isomorphisms of homology groups
are topological.
By Lemma 4.1, for a nuclear ⊗̂-algebra A which is a DF -space, the following com-
plex (BcΩ̄A+, b̃ + B̃) is a complex of nuclear complete DF -spaces and continuous
linear operators. By Corollary 3.6, for complete nuclear DF -spaces the isomor-
phisms for homology groups in (iii)N are also topological. �
Proposition 4.3. [18, Proposition 3.1] Let A be a complete locally convex algebra.
Then, for any even integer N , say N = 2K, and the following assertions, we have
(i)N ⇒ (ii)N ⇒ (iii)N ⇒ (ii)N+1 and (ii)N ⇒ (iv)N :
(i)N H
naive(A) = {0} for all n ≥ N and H
bar(A) = {0} for all n ≥ N − 1;
(ii)N for all n ≥ N, HH
n(A) = {0};
(iii)N for all k ≥ K, up to isomorphism of linear spaces, HC
2k(A) = HCN(A) and
HC2k+1(A) = HCN−1(A);
(iv)N up to isomorphism of linear spaces, HP
0(A) = HCN(A) and HP 1(A) =
HCN−1(A).
For nuclear Fréchet algebras the isomorphisms in (iii)N and (iv)N are topological
isomorphisms. For a nuclear ⊗̂-algebra A which is a DF -space the isomorphisms
in (iii)N are topological isomorphisms.
Proof. We need to add to the proof of [18, Proposition 3.1] the following part on
automatic continuity. In view of the proof of [18, Proposition 3.1] it is easy to
see that the isomorphisms of cohomology groups in (iii)N and (iv)N are induced by
continuous morphisms of complexes.
For nuclear Fréchet algebras, by Lemma 4.1, the complexes ((BcΩ̄A+)
∗, b̃∗ + B̃∗)
and ((BpΩ̄A+)
∗, b̃∗ + B̃∗) are complexes of strong duals of nuclear Fréchet spaces.
By Lemma 3.5, the isomorphisms of cohomology groups in (iii)N and (iv)N are
topological.
Cyclic cohomology of nuclear Fréchet and DF algebras 11
For a nuclear ⊗̂-algebra A which is a DF -space, by Lemma 4.1 and by [21,
Theorem 4.4.13], the chain complex (BcΩ̄A+, b̃+ B̃) is the strong dual of a complex
of nuclear Fréchet spaces. By [21, Theorem 4.4.12], complete nuclear DF -spaces and
nuclear Fréchet spaces are reflexive. Therefore ((BcΩ̄A+)
∗, b̃∗ + B̃∗) is a complex of
nuclear Fréchet spaces. Thus, by [7, Lemma 0.5.9], the isomorphisms of cohomology
groups in (iii)N are topological. �
The space of continuous traces on a topological algebra A is denoted by Atr, that
Atr = {f ∈ A∗ : f(ab) = f(ba) for all a, b ∈ A}.
The closure in A of the linear span of elements of the form {ab− ba : a, b ∈ A} is
denoted by [A,A]. Recall that b0 : A⊗̂A → A is uniquely determined by a ⊗ b 7→
ab− ba.
Proposition 4.4. Let A be in Φ and be a ⊗̂-algebra.
(i) Suppose that the continuous cohomology groups Hnaiven (A) = {0} for all n ≥ 1
and Hbarn (A) = {0} for all n ≥ 0. Then, up to isomorphism of linear spaces,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/Im b0;
HC2ℓ(A) = A/Im b0 and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
HP0(A) = A/Im b0 and HP1(A) = {0}.
(ii) Suppose that the continuous cohomology groups Hnnaive(A) = {0} for all n ≥ 1
and Hnbar(A) = {0} for all n ≥ 0. Then, up to isomorphism of linear spaces,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr;
HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
HP0(A) = Atr and HP1(A) = {0}.
Proof. (i). One can see that Hbarn (A) = {0} for all n ≥ 0 implies that
HHn(A) = H
naive
n (A) for all n ≥ 0,
see [17, Section 3]. Note that by definition of the ‘naive’ Hochschild homology of
A, Hnaive0 (A) = A/Im b0. Therefore, HHn(A) = {0} for all n ≥ 1 and HH0(A) =
A/Im b0.
From the exactness of the long Connes-Tsygan sequence of continuous homology
it follows that
HC0(A) = H
naive
0 (A) = A/Im b0 and HC1(A) = {0}.
The rest of Statement (i) follows from Proposition 4.2.
(ii) It is known that Hnbar(A) = {0} for all n ≥ 0, implies HH
n(A) = Hnnaive(A)
for all n ≥ 0. By definition of the ‘naive’ Hochschild cohomology of A, H0naive(A) =
Atr. Thus HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr.
From the exactness of the long Connes-Tsygan sequence of continuous cohomology
it follows that HC0(A) = H0naive(A) = A
tr and HC1(A) = {0}. The rest of
Statement (ii) follows from Proposition 4.3. �
12 Z. A. Lykova
5. Cyclic-type cohomology of biflat ⊗̂-algebras
Recall that a ⊗̂-algebra A is said to be biflat if it is flat in the category of A-⊗̂-
bimodules [7, Def. 7.2.5]. A ⊗̂-algebra A is said to be biprojective if it is projective
in the category of A-⊗̂-bimodules [7, Def. 4.5.1]. By [7, Proposition 4.5.6], a ⊗̂-
algebra A is biprojective if and only if there exists an A-⊗̂-bimodule morphism
ρA : A → A⊗̂A such that πA ◦ ρA = idA, where πA is the canonical morphism
πA : A⊗̂A → A, a1 ⊗ a2 7→ a1a2. It can be proved that any biprojective ⊗̂-algebra
is biflat and A = A2 = Im πA [7, Proposition 4.5.4]. Here A2 is the closure of
the linear span of the set {a1 · a2 : a1, a2 ∈ A} in A. A ⊗̂-algebra A is said to be
contractible if A+ is is projective in the category of A-⊗̂-bimodules. A ⊗̂-algebra A
is contractible if and only if A is biprojective and has an identity [7, Def. 4.5.8]. For
biflat Banach algebras A, Helemskii proved A = A2 = Im πA [7, Proposition 7.2.6]
and gave the description of the cyclic homology HC∗ and cohomology HC
∗ groups
of A in [8]. Later the author generalized Helemskii’s result to inverse limits of biflat
Banach algebras [17, Theorem 6.2] and to locally convex strict inductive limits of
amenable Banach algebras [18, Corollary 4.9].
Proposition 5.1. Let A be in Φ and be a biflat ⊗̂-algebra such that A = Im πA;
in particular, let A ∈ Φ be a biprojective ⊗̂-algebra. Then
(i) Hnaiven (A) = {0} for all n ≥ 1, H
naive
0 (A) = A/Im b0 and H
n (A) = {0} for all
n ≥ 0;
(ii) for the homology groups HH∗, HC∗ and HP∗ of A we have the isomorphisms of
linear spaces (5).
If, furthermore, A is a Fréchet space or A is a nuclear DF -space, then Hnaive0 (A) =
A/[A,A] is Hausdorff, and, for a biflat A, A = A2 implies that A = Im πA.
Proof. By [7, Theorem 3.4.25], up to topological isomorphism, the homology groups
Hnaiven (A) = Hn(A,A) = Tor
n (A,A+)
for all n ≥ 0. Since A is biflat, by [7, Proposition 7.1.2], Hnaiven (A) = {0} for all
n ≥ 1.
By [7, Theorem 3.4.26], up to topological isomorphism, the homology groups
Hbarn (A) = Hn+1(A,C) = Tor
n+1(C,C)
for all n ≥ 0, where C is the trivial A-bimodule. Note that, for the trivial A-
bimodule C, there is a flat resolution
0← C← A+ ← A← 0
in the category of left or right A-⊗̂-modules. By [7, Theorem 3.4.28], Hbarn (A) =
TorAn+1(C,C) = {0} for all n ≥ 1. By assumption, A = Im πA, hence H
0 (A) =
A/Im πA = {0}. Thus the conditions of Proposition 4.4 (i) are satisfied.
In the categories of Fréchet spaces and complete nuclear DF -spaces, the open
mapping theorem holds – see Corollary 3.4 for DF -spaces. Thus, by [7, Proposi-
tions 3.3.5 and 7.1.2], up to topological isomorphism, Hnaive0 (A) = Tor
0 (A,A+) is
Cyclic cohomology of nuclear Fréchet and DF algebras 13
Hausdorff. Since A is biflat, by [7, Proposition 7.1.2], TorA0 (C,A) is also Hausdorff.
By [7, Proposition 3.4.27], A2 = Im πA. �
A ⊗̂-algebra A is amenable if A+ is a flat A-⊗̂-bimodule. For a Fréchet algebra A
amenability is equivalent to the following: for all Fréchet A-bimodules X , H0(A, X)
is Hausdorff and Hn(A, X) = {0} for all n ≥ 1. Recall that an amenable Banach
algebra A is biflat and has a bounded approximate identity [7, Theorem VII.2.20].
Lemma 5.2. Let A be an amenable ⊗̂-algebra which is a Fréchet space or a nu-
clear DF -space. Then Hnaiven (A) = {0} for all n ≥ 1, H
naive
0 (A) = A/[A,A] and
Hbarn (A) = {0} for all n ≥ 0.
Proof. In the categories of Fréchet spaces and complete nuclear DF -spaces, the open
mapping theorem holds. Therefore, by [7, Theorem III.4.25 and Proposition 7.1.2],
up to topological isomorphism, for the trivial A-bimodule C,
Hbarn (A) = Hn+1(A,C) = Tor
n+1(C,A+) = {0}
for all n ≥ 0;
Hnaiven (A) = Hn(A,A) = Tor
n (A,A+) = {0}
for all n ≥ 1 and Hnaive0 (A) = Tor
0 (A,A+) is Hausdorff, that is, H
naive
0 (A) =
A/[A,A]. �
Theorem 5.3. Let A be a ⊗̂-algebra which is a Fréchet space or a nuclear DF -
space. Suppose that the continuous homology groups Hnaiven (A) = {0} for all n ≥ 1,
Hnaive0 (A) is Hausdorff and H
n (A) = {0} for all n ≥ 0. In particular, asssume
that A is a biflat algebra such that A = A2 or A is amenable. Then
(i) up to topological isomorphism,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/[A,A];
HC2ℓ(A) = A/[A,A] and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
(ii) up to topological isomorphism for Fréchet algebras and up to isomorphism of
linear spaces for nuclear DF -algebras,
(8) HP0(A) = A/[A,A] and HP1(A) = {0};
(iii)
Hnnaive(A) = {0} for all n ≥ 1;
Hnbar(A) = {0}; for all n ≥ 0;
(iv) up to topological isomorphism for nuclear Fréchet algebras and nuclear DF -
algebras and up to isomorphism of linear spaces for Fréchet algebras,
HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr;
HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
(v) up to topological isomorphism for nuclear Fréchet algebras and up to isomorphism
of linear spaces for Fréchet algebras and for nuclear DF -algebras,
(11) HP0(A) = Atr and HP1(A) = {0}.
14 Z. A. Lykova
Proof. In view of Proposition 5.1 and Lemma 5.2, a biflat algebra A such that
A = A2 and an amenable A satisfy the conditions of the theorem.
By Proposition 2.2, firstly, Hbarn (A) = {0} for all n ≥ 0 if and only if H
bar(A) =
{0} for all n ≥ 0; and, secondly, Hnnaive(A) = {0} for all n ≥ 1 if and only if
Hnaiven (A) = {0} for all n ≥ 1 and H
naive
0 (A) is Hausdorff.
By Proposition 4.4, we have isomorphisms of linear spaces in (i) – (v). In Propo-
sitions 4.2 and 4.2 we show also when the above isomorphisms are automatically
topological. �
Remark 5.4. Recall that, for a biflat Banach algebra A, dbwA ≤ 2 [27, Theorem
6]. By [14, Theorem 5.2], for a Banach algebra A of a finite weak bidimension dbwA,
we have isomorphisms between the entire cyclic cohomology and the periodic cyclic
cohomology of A, HE0(A) = HP 0(A) = Atr and HE1(A) = HP 1(A) = {0}. The
entire cyclic cohomology HEk(A) of A for k = 0, 1 are defined in [2, IV.7]. In
[25, Theorem 6.1] M. Puschnigg extended M. Khalkhali’s result on the isomorphism
HEk(A) = HP k(A) for k = 0, 1 from Banach algebras to some Fréchet algebras.
The following statement shows that the above theorems give the explicit descrip-
tion of cyclic type homology and cohomology of the projective tensor product of two
biprojective ⊗̂-algebras.
Proposition 5.5. Let B and C be biprojective ⊗̂-algebras. Then the projective
tensor product A = B⊗̂C is a biprojective ⊗̂-algebra.
Proof. Since B is biprojective, there is a morphism of B-⊗̂-bimodules ρB : B → B⊗̂B
such that πB ◦ ρB = idB. A similar statement is valid for C. Let i be the topological
isomorphism
i : (B⊗̂B)⊗̂(C⊗̂C)→ (B⊗̂C)⊗̂(B⊗̂C)
given by (b1⊗ b2)⊗ (c1⊗ c2) 7→ (b1⊗ c1)⊗ (b2⊗ c2). Note that πB⊗̂C = (πB⊗̂πC)◦ i
It is routine to check that
ρB⊗̂C : B⊗̂C → (B⊗̂C)⊗̂(B⊗̂C)
defined by ρB⊗̂C = i◦(ρB⊗ρC) is a morphism of B⊗̂C-⊗̂-bimodules and πB⊗̂C◦ρB⊗̂C =
idB⊗̂C. �
Remark 5.6. For amenable Banach algebras B and C, B. E. Johnson showed that
the Banach algebra A = B⊗̂C is amenable [11]. By [19, Proposition 5.4], for a biflat
Banach algebra A, each closed two-sided ideal I with bounded approximate identity
is amenable and the quotient algebra A/I is biflat. Thus the explicit description of
cyclic type homology and cohomology of such I and A/I is also given in Theorem
5.3. One can find a number of examples of biflat and simplicially trivial Banach and
C∗- algebras in [17, Example 4.6, 4.9].
Cyclic cohomology of nuclear Fréchet and DF algebras 15
6. Applications to the cyclic-type cohomology of biprojective
⊗̂-algebras
In this section we present examples of nuclear biprojective ⊗̂-algebras which are
Fréchet spaces or DF -spaces and the continuous cyclic-type homology and cohomol-
ogy of these algebras.
Example 6.1. Let G be a compact Lie group and let E(G) be the nuclear Fréchet
algebra of smooth functions on G with the convolution product. It was shown by
Yu.V. Selivanov that A = E(G) is biprojective [29].
Let E∗(G) be the strong dual to E(G), so that E∗(G) is a complete nuclear DF -
space. This is a ⊗̂-algebra with respect to convolution multiplication: for f, g ∈
E∗(G) and x ∈ E(G), < f ∗ g, x >=< f, y >, where y ∈ E(G) is defined by
y(s) =< g, xs >, s ∈ G and xs(t) = x(s
−1t), t ∈ G. J.L. Taylor proved that the
algebra of distributions E∗(G) on a compact Lie group G is contractible [30].
Example 6.2. Let (E, F ) be a pair of complete Hausdorff locally convex spaces
endowed with a jointly continuous bilinear form 〈·, ·〉 : E ×F → C that is not iden-
tically zero. The space A = E⊗̂F is a ⊗̂-algebra with respect to the multiplication
defined by
(x1 ⊗ x2)(y1 ⊗ y2) = 〈x2, y1〉x1 ⊗ y2, xi ∈ E, yi ∈ F.
Yu.V. Selivanov proved that this algebra is biprojective and usually non unital
[28, 29]. More exactly, if A = E⊗̂F has a left or right identity, then E or F
respectively is finite-dimensional. If the form 〈·, ·〉 is nondegenerate, then A = E⊗̂F
is called the tensor algebra generated by the duality (E, F, 〈·, ·〉).
In particular, if E is a Banach space with the approximation property, then the
algebra A = E⊗̂E∗ is isomorphic to the algebra N (E) of nuclear operators on E [7,
II.2.5].
6.1. Köthe sequence algebras. The following results on Köthe algebras can be
found in A. Yu. Pirkovskii’s papers [22, 23].
A set P of nonnegative real-valued sequences p = (pi)i∈N is called a Köthe set if
the following axioms are satisfied:
(P1) for every i ∈ N there is p ∈ P such that pi > 0;
(P2) for every p, q ∈ P there is r ∈ P such that max{pi, qi} ≤ ri for all i ∈ N.
Suppose, in addition, the following condition is satisfied:
(P3) for every p ∈ P there exist q ∈ P and a constant C > 0 such that pi ≤ Cq
for all i ∈ N.
For any Köthe set P which satisfies (P3), the Köthe space
λ(P ) = {x = (xn) ∈ C
N : ‖x‖p =
|xn|pn <∞ for all p ∈ P}
is a complete locally convex space with the topology determined by the family of
seminorms {‖x‖p : p ∈ P} and a ⊗̂-algebra with pointwise multiplication. The
⊗̂-algebras λ(P ) are called Köthe algebras.
By [21] and [6], for a Köthe set, λ(P ) is nuclear if and only if
(P4) for every p ∈ P there exist q ∈ P and ξ ∈ ℓ1 such that pi ≤ ξiqi for all i ∈ N.
16 Z. A. Lykova
By [22, Theorem 3.5], λ(P ) is biprojective if and only if
(P5) for every p ∈ P there exist q ∈ P and a constant M > 0 such that p2i ≤Mqi
for all i ∈ N.
The algebra λ(P ) is unital if and only if
n pn <∞ for every p ∈ P .
Example 6.3. Fix a real number 1 ≤ R ≤ ∞ and a nondecreasing sequence
α = (αi) of positive numbers with limi→∞ αi =∞. The power series space
ΛR(α) = {x = (xn) ∈ C
N : ‖x‖r =
|xn|r
αn <∞ for all 0 < r < R}
is a Fréchet Köthe algebra with pointwise multiplication. The topology of ΛR(α) is
determined by a countable family of seminorms {‖x‖rk : k ∈ N} where {rk} is an
arbitrary increasing sequence converging to R.
By [23, Corollary 3.3], ΛR(α) is biprojective if and only if R = 1 or R =∞.
By the Grothendieck-Pietsch criterion, ΛR(α) is nuclear if and only if for limn
0 for R <∞ and limn
<∞ for R =∞, see [22, Example 3.4].
The algebra ΛR((n)) is topologically isomorphic to the algebra of functions holo-
morphic on the open disc of radius R, endowed with Hadamard product, that is,
with “co-ordinatewise” product of the Taylor expansions of holomorphic functions.
Example 6.4. The algebra H(C) ∼= Λ∞((n)) of entire functions, endowed with the
Hadamard product, is a biprojective nuclear Fréchet algebra [23].
Example 6.5. The algebra H(D1) ∼= Λ1((n)) of functions holomorphic on the open
unit disc, endowed with the Hadamard product, is a biprojective nuclear Fréchet
algebra. Moreover it is contractible, since the function z 7→ (1− z)−1 is an identity
for H(D1) [23].
For any Köthe space λ(P ) the dual space λ(P )∗ can be canonically identified with
{(yn) ∈ C
N : ∃p ∈ P and C > 0 such that |yn| ≤ Cpn for all n ∈ N}.
It is shown in [23] that, for a biprojective Köthe algebra λ(P ), λ(P )∗ is a sequence
algebra with pointwise multiplication.
The algebra λ(P )∗ is unital if and only if there exists p ∈ P such that inf i pi > 0.
Example 6.6. The nuclear Fréchet algebra of rapidly decreasing sequences
s = {x = (xn) ∈ C
N : ‖x‖k =
|xn|n
k <∞ for all k ∈ N}
is a biprojective Köthe algebra [22]. The algebra s is topologically isomorphic to
Λ∞(α) with αn = logn [23]. The nuclear Köthe ⊗̂-algebra s
∗ of sequences of poly-
nomial growth is contractible [30].
Example 6.7. [23, Section 4.2] Let P be a Köthe set such that pi ≥ 1 for all p ∈ P
and all n ∈ N. Then the formula 〈a, b〉 =
i aibi defines a jointly continuous,
nondegenerate bilinear form on λ(P ) × λ(P ). Thus M(P ) = λ(P )⊗̂λ(P ) can be
considered as the tensor algebra generated by the duality (λ(P ), λ(P ), 〈·, ·〉), and so
is biprojective. There is a canonical isomorphism between M(P ) and the algebra
Cyclic cohomology of nuclear Fréchet and DF algebras 17
λ(P × P ) of N×N complex matrices (aij)(ij)∈N×N satisfying the condition ‖a‖p =
i,j |aij|pipj <∞ for all p ∈ P with the usual matrix multiplication.
In particular, for P = {(nk)n∈N : k = 0, 1, . . . }, we obtain the biprojective nuclear
Fréchet algebra ℜ = s⊗̂s of “smooth compact operators” consisting of N×N com-
plex matrices (aij) with rapidly decreasing matrix entries. Here s is from Example
Theorem 6.8. Let A be a ⊗̂-algebra belonging to one of the following classes:
(i) A = E(G) or A = E∗(G) for a compact Lie group G;
(ii) A = E⊗̂F , the tensor algebra generated by the duality (E, F, 〈·, ·〉) for nuclear
Fréchet spaces E and F (e.g., ℜ = s⊗̂s) or for nuclear complete DF -spaces E and
(iii) Fréchet Köthe algebras A = λ(P ) such that the Köthe set P satisfies (P3),
(P4) and (P5); in particular, Λ1(α) such that limn
= 0 or Λ∞(α) such that
<∞. (e.g., H(D1), s, H(C)).
(iv) Köthe algebras A = λ(P )∗ which are the strong duals of λ(P ) from (iii).
(v) the projective tensor product A = B⊗̂C of biprojective nuclear ⊗̂-algebras B
and C which are Fréchet spaces or DF -spaces; in particular, A = E(G)⊗̂ℜ.
Then, up to topological isomorphism,
Hnaiven (A) = {0} for all n ≥ 1 and H
naive
0 (A) = A/[A,A];
Hbarn (A) = {0} for all n ≥ 0;
HHn(A) = {0} for all n ≥ 1 and HH0(A) = A/[A,A];
HC2ℓ(A) = A/[A,A] and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
Hnnaive(A) = {0} for all n ≥ 1; H
bar(A) = {0} for all n ≥ 0;
HHn(A) = {0} for all n ≥ 1 and HH0(A) = Atr;
HC2ℓ(A) = Atr and HC2ℓ+1(A) = {0} for all ℓ ≥ 0;
and, up to topological isomorphism for Fréchet algebras and up to isomorphism of
linear spaces for DF -algebras,
HP0(A) = A/[A,A] and HP1(A) = {0};
HP0(A) = Atr and HP1(A) = {0}.
Proof. We have mentioned above that the algebras in (i)-(iii) and (v) are biprojective
and nuclear. By [23, Corollary 3.10], for any nuclear biprojective Fréchet Köthe
algebra λ(P ), the strong dual λ(P )∗ is a nuclear, biprojective Köthe ⊗̂-algebra
which is a DF -space. For nuclear Fréchet algebras and for nuclear DF -algebras, the
conditions of Theorem 5.3 are satisfied. Therefore, for the homology and cohomology
groups HH and HC of A we have the topological isomorphisms (7) and (10). For
the periodic cyclic homology and cohomology groups HP of A, for Fréchet algebras,
we have topological isomorphisms and, for nuclear DF -algebras, isomorphisms of
linear spaces (8) and (11). It is obvious that, for commutative algebras, Atr = A∗
and A/[A,A] = A. �
18 Z. A. Lykova
The cyclic-type homology and cohomology of E(G) for a compact Lie group G
were calculated in [20].
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School of Mathematics and Statistics, University of Newcastle,, Newcastle
upon Tyne, NE1 7RU, UK (Z.A.Lykova@newcastle.ac.uk)
	1. Introduction
	2. Definitions and notation
	3. The open mapping theorem in complete nuclear DF-spaces
	4. Cyclic and Hochschild cohomology of some -algebras
	5. Cyclic-type cohomology of biflat -algebras
	6. Applications to the cyclic-type cohomology of biprojective -algebras
	6.1.  Köthe sequence algebras
	References
