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Topological amenability and K\"othe co-echelon algebras

We introduce a notion of a topologically flat locally convex module, which extends the notion of a flat Banach module and which is well adapted to the nonmetrizable setting (and especially to the setting of DF-modules). By using this notion, we introduce topologically amenable locally convex algebras and we show that a complete barrelled DF-algebra is topologically amenable if and only if it is Johnson amenable, extending thereby Helemskii-Sheinberg's criterion for Banach algebras. As an application, we completely characterize topologically amenable K\"othe co-echelon algebras.


Introduction
The paper is devoted to the study of amenability properties in the framework of DF-algebras. These are algebras with jointly continuous multiplication whose underlying topological vector spaces are DF-spaces. The category of DF-spaces contains spaces of distributions, e.g. tempered distributions or distributions with compact support. More generally, duals of Fréchet spaces belong to this category. In particular, the duals of Köthe echelon spaces are DF-spaces. These are the so-called Köthe co-echelon spaces and this class of objects will be of particular importance for us.
The general study of amenable DF-algebras meets two major difficulties which come from the facts that the category of DF-spaces does not respect subspaces and that there is no Open Mapping Theorem available. This implies that the two wellknown approaches to amenability (namely, Johnson's approach based on derivations [10] and Helemskii-Sheinberg's approach based on flat modules [6]) which are equivalent in the category of Banach (or Fréchet) algebras are potentially inequivalent in the DF-algebra framework (however, we have no explicit counterexample so far). The main aim of this paper is to modify the notion of a flat module in such a way that the above-mentioned problem disappears. The resulting notion of a topologically flat module is equivalent to that of a flat module in the case of Banach (or Fréchet) modules, but, in our view, is better adapted to the nonmetrizable setting. We define topologically amenable algebras in terms of topologically flat modules, and we show that topological amenability for complete barrelled DF-algebras is equivalent to amenability in Johnson's sense. We also obtain a topological amenability criterion for Köthe co-echelon algebras, completing thereby recent results of the second author [23,24].
The theory of amenable Banach algebras essentially starts with the famous result of Johnson [10,Theorem 2.5] who proved that the convolution algebra L 1 (G) is amenable if and only if the locally compact group G is amenable. Since then amenable Banach algebras became an inseparable part of functional analysis and operator algebra theory (see [28] for a recent and detailed account). A few years after the publication of Johnson's memoir, Helemskii and Sheinberg [6] observed that the notion of an amenable algebra perfectly fits into the general "Banach homological algebra" developed earlier by Helemskii [5] (and, independently, by Kiehl and Verdier [11] and by Taylor [31]). Namely, Helemskii and Sheinberg proved that a Banach algebra A is amenable in Johnson's sense if and only if the unitization of A is a flat Banach A-bimodule. This result was extended by the first author [20,Corollary 3.5] to the setting of Fréchet algebras. In the present article we continue this investigation and study amenability properties of DF-algebras, with a special emphasis on Köthe co-echelon algebras.
The paper is organized as follows. The next section is Notation and Preliminaries, and it contains basic definitions, facts and notation that is used in the sequel. In Section 3, we introduce and study topologically flat locally convex modules and topologically amenable locally convex algebras. The main results here are Theorem 3.12, which characterizes topologically flat DF-modules in terms of the Ext functor, and Theorem 3.18, which shows that, for complete barrelled DF-algebras, the topological amenability in our sense is equivalent to the Johnson amenability. In Section 4, we characterize topologically amenable Köthe co-echelon algebras k p (V ) in terms of the corresponding weight sets V (Theorems 4.4 and 4.5). Finally, in Section 5 we give some concrete examples of topologically amenable (and nonamenable) co-echelon algebras. In particular, we construct a topologically amenable co-echelon algebra of order ∞ which, in a sense, cannot be reduced to a direct sum of ℓ ∞ with a contractible co-echelon algebra.
General references are: [17] for functional analysis, [4,15] for Banach and topological algebra theory, and [7] for the homology theory of topological algebras.

Notation and Preliminaries
We start by recalling some basic definitions and introducing some notation that will be used in the sequel. By a locally convex algebra we mean a locally convex space (lcs) over C equipped with a separately continuous associative multiplication. In general, locally convex algebras are not assumed to have an identity. Given a locally convex algebra A, we denote by A + the unconditional unitization of A, and we denote by A op the opposite algebra, i.e., the lcs A with multiplication a · b := ba. In what follows, when using the word "algebra" with an adjective that describes a linear topological property (such as "complete", "Fréchet", "Banach", etc.), we mean that the underlying lcs of the algebra in question has the specified property. The same applies to locally convex modules (see below).
Given a locally convex algebra A, a left locally convex A-module is an lcs X together with a left A-module structure such that the action A×X → X is separately continuous. Right locally convex modules and locally convex bimodules are defined similarly. At some point we will be using a concrete locally convex bimodule A ⊗ C which is the lcs A itself with trivial right module action and multiplication as the left module action.
The completed projective tensor product of lcs's E and F will be denoted by E ⊗ F , and the completion of E will be denoted by E or by E ∼ . A complete locally convex algebra with jointly continuous multiplication is called a ⊗-algebra. If A is a ⊗-algebra then the assignment a ⊗ b → ab gives rise to the so-called product map π A : A ⊗ A → A. We will simply write π whenever it is clear to which algebra the product map is referred to. If A is a ⊗-algebra, then a left locally convex A-module X is a left A-⊗-module if X is complete and if the action of A on X is jointly continuous. Right ⊗-modules and ⊗-bimodules are defined similarly. The category of left A-⊗-modules (respectively, of right A-⊗-modules, of A-B-⊗-bimodules) will be denoted by A-mod (respectively, mod-A, A-mod-B). Note that A-⊗-bimodules are nothing but left unital A e -⊗-modules, where A e := A + ⊗ A op + is the enveloping algebra of A (see [7,§II.5.2]). If X ∈ mod-A and Y ∈ A-mod, then the A-module projective tensor product of X and Y is defined as If X and Y are two lcs's, then L(X, Y ) stands for the vector space of continuous linear operators from X to Y . We equip L(X, Y ) with the topology of uniform convergence on bounded sets. As usual, we let X ′ = L(X, C). If A is a locally convex algebra and X, Y are left locally convex A-modules, then A L(X, Y ) denotes the vector space of continuous linear A-module maps, i.e., operators T ∈ L(X, Y ) satisfying T (a · x) = a · T x for all a ∈ A, x ∈ X. In the case of right A-modules, resp. A-B-bimodules, the vector spaces L A (X, Y ) and A L B (X, Y ) are defined analogously.
Suppose that A, B, C are locally convex algebras, X is a locally convex B-Cbimodule, and Y is a locally convex A-C-bimodule. Then L C (X, Y ) has a natural A-B-bimodule structure given by If the actions of A on Y and of B on X are hypocontinuous with respect to the families of bounded subsets of Y and X, respectively, then L C (X, Y ) is easily seen to be a locally convex A-B-bimodule (cf. [31,Section 3]). In particular, this condition is satisfied provided that the actions are jointly continuous. In particular, for each ⊗-algebra A and each left (respectively, right) A-⊗-module X the dual space X ′ is a right (respectively, left) locally convex A-module. Note, however, that the action of A on X ′ need not be jointly continuous. Let CLCS denote the category of complete lcs's and continuous linear maps. Suppose that C ⊂ CLCS is a full additive subcategory. We write alg(C) for the category of all ⊗-algebras whose underlying spaces are objects of C. If A is a ⊗algebra, then we denote by A-mod(C) the full subcategory of A-mod consisting of those modules whose underlying spaces are objects of C. The symbols mod-A(C) and A-mod-B(C) are understood in a similar way.
Following [22] (cf. also [7]), we say that C is admissible if the following holds: (C1) if E ∈ C and F is a locally convex space isomorphic to E, then F ∈ C; (C2) if E ∈ C and E 0 ⊂ E is a complemented vector subspace, then E 0 ∈ C; Most of the categories of complete lcs's used in functional analysis are admissible. In this paper, the concrete admissible subcategories we are mostly interested in are CLCS itself, the category Ban of Banach spaces, the category Fr of Fréchet spaces, and the category CBDF of complete barrelled (DF)-spaces. The admissibility of Ban and Fr is well known. As for CBDF, property (C2) follows from the fact that the classes of barrelled spaces and of (DF)-spaces are stable under taking quotients modulo closed subspaces [12, 27. Let A be a ⊗-algebra, and let C be an admissible subcategory of CLCS. A sequence , if it has a contracting homotopy consisting of continuous linear maps. Geometrically, this means that i is a topological embedding, p is open (i.e., is a quotient map), i(X) = ker p, and i(X) is a complemented subspace of Y . We say that a morphism i : X → Y (respectively, p : Y → Z) in A-mod(C) is an admissible monomorphism (respectively, an admissible epimorphism) if it fits into an admissible sequence (1). It is easy to show that A-mod(C) together with the class of all admissible sequences is an exact category in Quillen's sense [26]. Therefore most of the main notions and constructions of homological algebra (projective objects, projective resolutions, derived functors, etc.) make sense in A-mod(C). For details, we refer to [7]. An important property of A-mod(C) is that, if A ∈ alg(C), then A-mod(C) has enough projectives. As a consequence, each covariant functor F : A-mod(C) → Vect (where Vect is the category of vector spaces) has left derived functors L n F , and each contravariant functor F : A-mod(C) → Vect has right derived functors R n F (n ≥ 0). In particular, for each left locally convex A-module Y the functor Ext n A (−, Y ) is defined to be the nth right derived functor of A L(−, Y ) : A-mod(C) → Vect. We would like to stress that, in contrast to [7], we do not require Y to be an object of A-mod(C). In particular, we may let Y = Z ′ for some Z ∈ mod-A(C). This special case will be essential in our characterization of topologically flat modules (see Theorem 3.12). In fact, this is the only reason why we have to consider general locally convex modules rather than ⊗-modules only.
Note that the above facts on A-mod(C) have obvious analogs for mod-A(C) and A-mod-B(C). For details, see [7].
Let us now recall some facts on strictly exact sequences of locally convex spaces. Let C be an additive category. Following [30], we say that a short sequence (1) in C is strictly exact if i is a kernel of p and p is a cokernel of i.
We end this section with a definition and a collection of basic facts concerning Köthe co-echelon spaces and algebras. Let I be a countable set, and let V : We often write k p (V ) for k p (I, V ) when the index set I is clear from the context.
In most examples we actually have I = N (see Examples 2.5-2.8), but sometimes it is more convenient to let I = N × N (see Example 5.6). The above definition is a bit unusual since we allow v n (i) = ∞ for some n ∈ N and i ∈ I. However, this less restrictive approach does not affect our proofs and allows us to consider in particular the space ϕ := C (N) of finitely supported sequences (see Example 2.5 below). The space k p (I, V ) is canonically endowed with the inductive limit topology of the system (ℓ p (I, v n )) n∈N (for p ≥ 1) or (c 0 (I, v n )) n∈N (for p = 0), where ℓ p (I, v n ) and c 0 (I, v n ) are the weighted Banach spaces of scalar sequences equipped with their canonical norms. Clearly, if v n (i) = ∞, then x ∈ ℓ p (I, v n ) implies that x i = 0. Thus we usually write Since Köthe co-echelon spaces are countable inductive limits of Banach spaces, they are barrelled DF-spaces (see [9,12.4,Theorem 8] (we let ∞/∞ = 1 for convenience). Moreover, if (W3) holds, then the multiplication on k p (V ) is automatically jointly continuous [loc. cit.]. From now on, when we write something like "let k p (V ) be a Köthe co-echelon algebra", we tacitly assume that V is a sequence of weights satisfying conditions (W1)-(W3), and that k p (V ) is considered as a locally convex algebra under the coordinatewise multiplication.
and v n (j) = ∞ for j > n. Conditions (W1)-(W3) are clearly satisfied, and k p (V ) is nothing but the algebra ϕ of finite sequences equipped with the strongest locally convex topology.
Example 2.6. Let R ∈ [0, +∞), and let α = (α i ) i∈N be a sequence of positive numbers increasing to infinity. Consider the dual power series spaces 1 If (r n ) is a fixed sequence of positive numbers strictly decreasing to R, then we Equivalently, this means that if r > R, then r 2 > R. If R ≥ 1 or R = 0, then this condition is clearly satisfied, so DΛ p R (α) is a Köthe co-echelon algebra in this case. If 0 < R < 1, then the above condition fails (take any r ∈ (R, √ R]).
Example 2.7. Letting α j = log j in Example 2.6, we see that DΛ p 0 (α) is nothing but the algebra s ′ of sequences of polynomial growth.
Example 2.8. If α j = j, then DΛ p R (α) is topologically isomorphic to the space of germs of holomorphic functions on the closed disc D R = {z ∈ C : |z| ≤ R}. If R ≥ 1 or R = 0, then the multiplication on DΛ p R (α) corresponds to the "componentwise" multiplication of the Taylor expansions of holomorphic functions (the Hadamard multiplication, cf. [27]). The resulting locally convex algebra will be denoted by H (D R ).
In fact, if the above conditions are satisfied, then we have [2,Proposition 15] A comprehensive study of Köthe co-echelon spaces may be found in [1]. Köthe co-echelon algebras appear as a main object of investigation in [3] and [23,24].

Topological Flatness and Topological Amenability
Let C be an admissible subcategory of CLCS, and let A ∈ alg(C).
Remark 3.2. According to [7], a module X ∈ A-mod(C) is flat (relative to C) if for each short admissible sequence (2) in mod-A(C) the sequence (3) is exact in Vect.
If C ⊂ Fr, then flatness and topological flatness are equivalent (see Example 2.3). We conjecture that, in the general case, neither topological flatness implies flatness, nor vice versa. However, we do not have concrete counterexamples at the moment.
which is split exact and is a fortiori strictly exact in CLCS. Since each projective module is a retract of a free module [7, III.1.27], the result follows.
Proof. This is immediate from Definition 3.1 and from the fact that the functor (−) ⊗ A X : mod-A → CLCS preserves cokernels [21,Proposition 3.3]. For C = Fr, this fact was observed in [20].
The following "adjoint associativity" (or "exponential law") for locally convex spaces is a kind of folklore. Since we have not found an exact reference, we give a proof here for the convenience of the reader. Proposition 3.6. Let X, Y , Z be locally convex spaces. Suppose that Z is complete. There is a natural linear map The above map is a vector space isomorphism in either of the following cases: (i) X and Y are Fréchet spaces; (ii) X and Y are DF-spaces, and Y is barrelled.
Proof. By the universal property of the projective tensor product (see, e.g., [13, 41.3. (1)]), L(X ⊗ Y, Z) is naturally identified with the space of jointly continuous bilinear maps from X × Y to Z. On the other hand, each ϕ ∈ L(X, L(Y, Z)) determines a separately continuous bilinear map Φ : . Moreover, the rule ϕ → Φ determines a vector space isomorphism between L(X, L(Y, Z)) and the space of those separately continuous bilinear The following is a natural extension of [7,II.5.22] to the locally convex setting.
Proposition 3.8. Let A, B, C be ⊗-algebras, and let X ∈ A-mod-B, Y ∈ B-mod-C, and Z ∈ A-mod-C. There is a natural linear map The Corollary 3.10. Let C ∈ {Fr, CBDF}, let A be a ⊗-algebra, and let X ∈ mod-A(C), Y ∈ A-mod(C), Z ∈ C. Then there exists a natural vector space isomorphism Proof. This follows from Corollaries 3.7, 3.9, the commutativity of ⊗, and the associativity of ⊗ A , since we have The following result was proved in [20, Proposition 3.3] for C = Fr. We now give a shorter proof which holds both for C = Fr and C = CBDF.
Proposition 3.11. Let C ∈ {Fr, CBDF}, and let A ∈ alg(C). Then for all X ∈ A-mod(C), Y ∈ mod-A(C), n ∈ Z + there is a natural vector space isomorphism Applying Corollary 3.9 twice, we obtain natural vector space isomorphisms Our next theorem generalizes [20, Proposition 3.4].
Theorem 3.12. Let C ∈ {Fr, CBDF}, and let A ∈ alg(C). The following properties of X ∈ A-mod(C) are equivalent: is exact in Vect. Corollary 3.9 implies that (6) is isomorphic to This yields (vi). (vi) ⇒ (i). We want to show that for each short admissible sequence (2) in mod-A(C) the sequence (3) is strictly exact in CLCS. By Palamodov's Theorem 2.4, this means precisely that for each set S the sequence is exact in Vect. Taking into account the isomorphism ℓ ∞ (S) ∼ = (ℓ 1 (S)) ′ and applying Corollary 3.10, we see that (8) is isomorphic to Since ℓ 1 (S) ⊗ Y • is admissible in mod-A(C), we see that (9) is exact in Vect by (vi). In view of the above remarks, this completes the proof.
The next proposition shows that a flat Banach module over a Banach algebra remains topologically flat if we consider it as an object of the bigger category of Fréchet modules or of complete barrelled DF-modules. Proposition 3.13. Let A be a Banach algebra and let X be a left Banach A-module. The following conditions are equivalent: (i) X is flat (or, equivalently, topologically flat) relative to Ban; (ii) X is flat (or, equivalently, topologically flat) relative to Fr; (iii) X is topologically flat relative to CBDF.
Proof. Clearly, each of the conditions (ii) and (iii) implies (i). Conversely, let C denote either of the categories Fr or CBDF, and suppose that (i) holds. By [7, VII.1.14], condition (i) means precisely that X ′ is injective in mod-A(Ban). Using [7, III.1.31], we see that X ′ is a retract of L(A + , X ′ ) in mod-A(Ban). Hence for each short admissible sequence Y Applying Theorem 2.4, we conclude that X is topologically flat in A-mod(C).
We now turn to topological amenability, using Helemskii-Sheinberg's approach [6] as a motivation. Let C be an admissible subcategory of CLCS, and let A ∈ alg(C). Definition 3.15. We say that A is topologically amenable (relative to C) if A + is topologically flat in A-mod-A(C).
Remark 3.16. According to [7], A is amenable if A + is flat in A-mod-A(C). As in Remark 3.2, we would like to stress that amenability and topological amenability are formally different in the general case, but they are equivalent if C ⊂ Fr.  [7,Postscript]) that A is contractible if A + is projective in A-mod-A(C). Since projective modules are topologically flat (see Example 3.3), we conclude that each contractible algebra is topologically amenable.
Recall that the amenability of a Banach algebra can be rephrased in the language of derivations. Our next result gives a similar characterization in the categories Fr and CBDF. For Fréchet algebras, this was proved in [20, Corollary 3.5].
Theorem 3.18. Let C ∈ {Fr, CBDF}, and let A ∈ alg(C). Then A is topologically amenable relative to C if and only if for each X ∈ A-mod-A(C) every continuous derivation A → X ′ is inner.
Proof. It is a standard fact (see, e.g., [7, Chap. I, Subsection 2.1]) that every continuous derivation A → X ′ is inner if and only if H 1 (A, X ′ ) = 0, where H 1 (A, X ′ ) is the 1st continuous Hochschild cohomology group of A with coefficients in X ′ . By [7,III.4.9], we have a vector space isomorphism H 1 (A, X ′ ) ∼ = Ext 1 A e (A + , X ′ ). Now the result follows from Theorem 3.12.
In the CBDF category it is also possible to relate topological amenability to amenability.
Corollary 3.19. Let A be a complete barrelled DF-algebra which is amenable relative to CBDF. Then A is topologically amenable relative to CBDF.
Proof. By [23,Theorem 4.4], for each X ∈ A-mod-A(CBDF) every continuous derivation A → X ′ is inner. Now the result follows from Theorem 3.18.
If A is a Banach algebra then the above notions coincide.
Proposition 3.20. Let C ∈ {Fr, CBDF}, and let A be a Banach algebra. Then A is topologically amenable relative to C if and only if A is amenable relative to Ban.
Proof. This follows immediately from Proposition 3.13.
Since the algebras k 0 (V ) that appear in the next section are not necessarily complete, we adopt the following definition of topological amenability for noncomplete algebras.
Definition 3.21. Let C ∈ {Fr, CBDF}, and let A be a locally convex algebra with jointly continuous multiplication such that A ∈ alg(C) (where A is the completion of A). We say that A is topologically amenable relative to C if A is topologically amenable relative to C.
Given A as above, let A-mod-A(C) denote the category of locally convex Abimodules X such that the left and right actions of A on X are jointly continuous and such that the underlying space of X is an object of C. Clearly, we have an isomorphism of categories A-mod-A(C) ∼ = A-mod-A(C).
Using the above definition, we can easily extend Theorem 3.18 to non-complete algebras.
Theorem 3.22. Let C ∈ {Fr, CBDF}, and let A be a locally convex algebra with jointly continuous multiplication such that A ∈ alg(C). Then A is topologically amenable relative to C if and only if for each X ∈ A-mod-A(C) every continuous derivation A → X ′ is inner.
Proof. Given X ∈ A-mod-A(C) ∼ = A-mod-A(C), observe that X ′ is complete (see, e.g., [12, 28.5. (1)]). Hence each continuous derivation A → X ′ uniquely extends to a continuous linear map A → X ′ , which is easily seen to be a derivation. Thus we have a 1-1 correspondence between the continuous derivations A → X ′ and A → X ′ , which takes the inner derivations onto the inner derivations. Now the result follows from Theorem 3.18 applied to A.
We end this section with another consequence of topological amenability. The proof is similar to that of [4,Proposition 2.8.64] therefore we omit it.
Proposition 3.23. Let C ∈ {Fr, CBDF}, and let A and B be locally convex algebras with jointly continuous multiplication such that A, B ∈ alg(C). Suppose that θ : A → B is a continuous homomorphism with dense range. If A is topologically amenable relative to C, then so is B.

Topological Amenability for Co-echelon Algebras
We are now going to investigate topological amenability in the framework of Köthe co-echelon algebras. Throughout this section, amenability and topological amenability are considered relative to the category CBDF of complete barrelled DF-spaces.
The following result is a restatement of [7, Lemma 0.5.1] adapted to DF-spaces. The proof is essentially the same.
Lemma 4.1. Let X and Y be DF-spaces such that X is complete and Y is quasibarrelled, and let u : X → Y be a continuous linear injection. If u has dense range and its adjoint u ′ : Y ′ → X ′ is surjective, then u is a topological isomorphism between X and Y .
Proof. By assumption, u ′ : Y ′ → X ′ is a continuous linear bijection between Fréchet spaces, thus it is a topological isomorphism by the Open Mapping Theorem [17,Theorem 24.30]. Therefore u ′′ is a topological isomorphism as well. We have where ι X : X ֒→ X ′′ and ι Y : Y ֒→ Y ′′ are the canonical inclusions. Since Y is quasi-barrelled, it follows from [9, 11.2, Proposition 2] that ι Y is a topological embedding. Since u ′′ is a topological isomorphism, we conclude from (10) that ι X is continuous, or, equivalently, a topological embedding [loc. cit.]. Hence u ′′ induces a topological isomorphism u : X → im u. Since X is complete, im u is complete as well, so im u is closed in Y . Therefore u is a topological isomorphism of X onto im u = im u = Y .
Before proceeding to the characterization results, we list some properties of topologically amenable Köthe co-echelon algebras of finite order.
. As a consequence, the quotient k p (V ) ⊗ k p (V )/ ker π is complete.
Proof. To begin with, let us show that the family (e i ⊗ e j ) i,j∈N is a Schauder basis in k p (V ) ⊗ k p (V ) with respect to the square ordering of N×N (see [29,Section 4.3]). Indeed, we have k p (V ) ⊗ k p (V ) = ind n ℓ p (v n ) ⊗ ℓ p (v n ) by [16,Theorem 7]. Hence if u ∈ k p (V ) ⊗ k p (V ) then u ∈ ℓ p (v n ) ⊗ ℓ p (v n ) for some n ∈ N. Since (e j ) j∈N is a Schauder basis in ℓ p (v n ), it follows from [29,Proposition 4.25] that (e i ⊗ e j ) i,j∈N is a Schauder basis in ℓ p (v n ) ⊗ ℓ p (v n ) with respect to the square ordering. Therefore x → x i on k p (V ) are obviously continuous, so are the functionals e * i ⊗ e * j on k p (V ) ⊗ k p (V ). Thus (e i ⊗ e j ) i,j∈N is a Schauder basis.
Given u = i,j u ij e i ⊗ e j ∈ k p (V ) ⊗ k p (V ), we clearly have π(u) = i u ii e i . Hence ker π = span{e i ⊗ e j : i = j}. Therefore, to complete the proof, it suffices to construct a continuous linear projection P on k p (V ) ⊗ k p (V ) such that P (e i ⊗ e j ) = δ ij e i ⊗ e j for all i, j, where δ ij is the Kronecker delta.
Given n ∈ N, let ℓ 0 p (v n ) denote the subspace of ℓ p (v n ) consisting of finite sequences. Consider the bilinear map x j y j e j ⊗ e j .
We claim that B n is bounded. Indeed, using [29, Lemma 2.22], we obtain where (r j ) are the Rademacher functions on [0, 1]. Hence Therefore B n is bounded. Extending B n by continuity to ℓ p (v n ) × ℓ p (v n ) and then linearizing, we obtain a bounded linear operator P n on ℓ p (v n ) ⊗ ℓ p (v n ). Finally, letting P = ind n P n , we obtain a continuous linear operator P on k p (V ) ⊗ k p (V ) with the required properties. In view of the above remarks, this completes the proof.
where q is the quotient map. Moreover, a j e j ⊗ e j + ker π (a ∈ k p (V )); Proof. (i) Suppose towards a contradiction that all the weights v n are unbounded. This implies that there is a sequence j l ր ∞ such that v k (j l ) ≥ 1 for all l ∈ N and all k ≤ l. Define a dense range homomorphism where we consider ℓ p with the coordinate-wise multiplication. For every k ∈ N we get θ(a) p ℓp = l≤k |a j l | p + l>k |a j l | p ≤ C k a p k,p with C k := max{1/v k (j l ) p : l ≤ k} + 1. Consequently, θ indeed takes k p (V ) to ℓ p and is continuous. Since k p (V ) is topologically amenable, it follows from Propositions 3.20 and 3.23 that the Banach algebra ℓ p is amenable. This leads to a contradiction since ℓ p is known to be non-amenable (see, e.g., [4, Example 4.1.42(iii)]). Therefore V is eventually bounded.
(ii) To prove that π is open, it suffices to show thatπ is a topological isomorphism. Taking into account Lemma 4.2, we see thatπ acts between complete barrelled DF-spaces and, clearly, has dense range. By Lemma 4.1, the proof will be complete if we show thatπ ′ is surjective. Towards this goal, take ψ ∈ (k p (V ) ⊗ k p (V )/ ker π) ′ and let ψ 0 = ψ • q. Since ψ 0 vanishes on ker π, we have Define now a linear map In other words, δ is the image of ψ 0 under (4) (where X = Y = k p (V ) and Z = C).
Since (e j ) j∈N is a Schauder basis in k p (V ), this implies (11).
(iii) To get the nuclearity of k p (V ) we repeat exactly the proof of [23, Theorem 5.1]. We can indeed do so, sinceπ −1 is a topological isomorphism not only in the case of amenability (which was the assumption in [23]) but also under the weaker assumption of topological amenability.  It turns out that the cases of Köthe co-echelon algebras of order zero and infinity can be treated simultaneously.
Theorem 4.5. Let p ∈ {0, ∞}, and let k p (V ) be a Köthe co-echelon algebra. TFAE: is topologically amenable then we can follow the proof of Proposition 4.3 to show that V is eventually bounded. Indeed, suppose towards a contradiction that all the weights v n are unbounded. This implies that there is a sequence j l ր ∞ such that v k (j l ) ≥ 2 l for all l ∈ N and all k ≤ l. Define a dense range homomorphism where we consider ℓ 1 with the coordinate-wise multiplication. For every k ∈ N we get with C k := l≤k (1/v k (j l )) + 1. Consequently, θ indeed takes k ∞ (V ) to ℓ 1 and is continuous. Since k ∞ (V ) is topologically amenable, it follows from Propositions 3.20 and 3.23 that the Banach algebra ℓ 1 is amenable. This leads to a contradiction since ℓ 1 is known to be non-amenable (see, e.g., [4, Example 4.1.42(iii)]). Therefore V is eventually bounded.
Consequently, b ε ∈ ℓ ∞ with b ε ℓ∞ ≤ 2C ε a 2 n,∞ . If J 2 is empty, we conclude that a = b ε is in the range of θ. Otherwise observe that For any j ∈ J 2 we get |a j |v m (j) ≤ C|a j |v n (j) 2 ≤ C a n,∞ v n (j) < C a n,∞ ε 2C a n,∞ = ε 2 < ε.
(i) ⇔ (iii). This part is even easier since (e j ) j∈N is a common Schauder basis for both c 0 and k 0 (V ), thus the density of the range of θ in (13)  The reader may have noticed that for all the algebras mentioned in Examples 5.1-5.4 topological amenability is equivalent to contractibility. On the other hand, there are two obvious examples of topologically amenable co-echelon algebras that are not contractible -namely, c 0 and ℓ ∞ . To construct more examples of the same kind, let us first observe that the direct sum of two co-echelon algebras of the same order is also a co-echelon algebra. More exactly, if V = (v n ) n∈N and W = (w n ) n∈N are sequences of weights on index sets I and J, respectively, then we have k p (I, V ) ⊕ k p (J, W ) ∼ = k p (I ⊔ J, U ), where the sequence U = (u n ) n∈N of weights on I ⊔ J is given by u n (i) = v n (i) if i ∈ I, and u n (j) = w n (j) if j ∈ J. Conversely, each partition I = S ⊔ T induces a direct sum decomposition k p (I, V ) ∼ = k p (S, V S ) ⊕ k p (T, V T ), where V S and V T consist of the restrictions to S and T of weights from V .
In view of the above discussion, A 1 and A 2 are co-echelon algebras of order 0 and ∞, respectively. By Theorem 4.5, A 1 and A 2 are topologically amenable. On the other hand, A 1 and A 2 are not Montel spaces, so they are not contractible by [24, Theorems 12 and 13] (moreover, A 1 is not unital, which already implies that it is not contractible).
Of course, the above example is degenerate in a sense. Our next goal is to construct a "genuine" example of a co-echelon algebra of order ∞ which is topologically amenable and unital, but is not contractible. By "genuine" we mean that the algebra we are going to construct is not reduced to a direct sum of ℓ ∞ with a contractible algebra of the form k ∞ (V ) in the sense explained before Example 5.5.
Example 5.6. We fix a sequence (c j ) j∈N of positive numbers such that c j ≤ 1 for all j, and such that c j → 0 as j → ∞. For each n ∈ N define a weight v n on N 2 by v n (i, j) = c n j , i < n, Clearly, the sequence V = (v n ) n∈N satisfies (W1) and (W2). Furthermore, we have v 2n ≤ v 2 n for all n ∈ N, whence V satisfies (W3). Thus k p (N 2 , V ) is a Köthe coechelon algebra for all p. Since V is eventually bounded, we see that k 0 (N 2 , V ) and k ∞ (N 2 , V ) are topologically amenable (see Theorem 4.5). Moreover, k ∞ (N 2 , V ) is clearly unital.
Lemma 5.7. If S ⊂ N 2 , then k ∞ (S, V S ) is a Banach space if and only if S ∩ L n is finite for all n ∈ N.
Proof. We will use the well-known fact that an (LB)-space E = ind n E n (where E n are Banach spaces, and E n → E n+1 are bounded linear injections) is a Banach space if and only if the sequence (E n ) stabilizes in the sense that there exists N ∈ N such that E n → E n+1 is a topological isomorphism for all n ≥ N (this follows, for example, from [12, 19.5

.(4)]).
If S ∩ L n is finite for all n ∈ N, then so is S n = k≤n (S ∩ L k ). We clearly have v n = v n+1 = 1 outside S n . Letting we obtain v n ≤ C n v n+1 everywhere on S. This readily implies that ℓ ∞ (S, v n ) → ℓ ∞ (S, v n+1 ) is a topological isomorphism. Hence k ∞ (S, V S ) is a Banach space. Conversely, suppose that S ∩ L k is infinite for some k. Since for each n ≥ k + 1 we have v n (k, j) = c n j , and since c j → 0 as j → ∞, we see that there is no C > 0 such that v n ≤ Cv n+1 on S ∩ L k . Therefore ℓ ∞ (S, v n ) → ℓ ∞ (S, v n+1 ) is not a topological isomorphism, and so k ∞ (S, V S ) is not a Banach space. Lemma 5.8. There is no decomposition N 2 = S ⊔ T such that k ∞ (S, V S ) is a Banach space and such that k ∞ (T, V T ) is a Montel space.
Proof. Suppose that S ⊂ N 2 is a subset such that k ∞ (S, V S ) is a Banach space, and let T = N 2 \ S. By Lemma 5.7, for each n ∈ N there exists j n ∈ N such that (n, j n ) ∈ T . We clearly have v m (n, j n ) = 1 for all n ≥ m. Letting R = {(n, j n ) : n ∈ N}, we conclude that The existence of an infinite set R ⊂ T with the above property means precisely that k ∞ (T, V T ) is not Montel [1,Theorem 4.7].
Essentially the same argument applies to k 0 (N 2 , V ). However, more is true.
Lemma 5.9. Let V be the weight sequence on N 2 given by (14). Then k 0 (N 2 , V ) is not complete, and the underlying lcs of k 0 (N 2 , V ) is not isomorphic to a direct sum of a normed space and a dense subspace of a reflexive space.
Proof. Recall from [1, Theorem 2.7] that, for each set I and each sequence V = (v n ) n∈N of weights on I satisfying (W1) and (W2), the strong dual of k 0 (I, V ) is topologically isomorphic to the Köthe echelon space where a n (i) = v n (i) −1 and A = (a n ) n∈N is the corresponding Köthe set. Let now I = N 2 , let V be given by (14), and let E = k 0 (N 2 , V ). Assume, towards a contradiction, that E ∼ = E 0 ⊕ E 1 , where E 0 is a normed space and E 1 is a dense subspace of a reflexive space. Hence we have a topological isomorphism E ′ ∼ = E ′ 0 ⊕ E ′ 1 . Moreover, E ′ 0 is a Banach space, and E ′ 1 is a reflexive Fréchet space (see, e.g., [12, 23.5.(5) and 29.3.(1)]). Now recall from [17,Corollary 25.14] that all reflexive Fréchet spaces are distinguished, i.e., their strong duals are barrelled. Clearly, each normed space is distinguished, and a direct sum of two distinguished spaces is distinguished. Therefore E ′ is distinguished.
On the other hand, it is easily seen that the Köthe set A = (a n ) n∈N on N 2 , where a n (i, j) = v n (i, j) −1 , satisfies the conditions of [17,Corollary 27.18]. Hence λ 1 (N 2 , A) is not distinguished. This is a contradiction since E ′ ∼ = λ 1 (N 2 , A) (see above).
Proposition 5.10. Let V be the weight sequence on N 2 given by (14). Then (i) k 0 (N 2 , V ) and k ∞ (N 2 , V ) are topologically amenable Köthe co-echelon algebras; (ii) k ∞ (N 2 , V ) is unital; (iii) k 0 (N 2 , V ) is not complete; (iv) there is no decomposition N 2 = S ⊔ T such that k ∞ (S, V S ) is a Banach algebra and such that k ∞ (T, V T ) is a contractible algebra; (v) the underlying lcs of k 0 (N 2 , V ) is not isomorphic to a direct sum of a normed algebra and a contractible Köthe co-echelon algebra.
Proof. Properties (i) and (ii) are mentioned in Example 5.6, while (iii) is contained in Lemma 5.9. To prove (iv) and (v), observe that each contractible co-echelon algebra of order p > 0 is a Montel space (for p < ∞ this follows from Theorem 4.4, while for p = ∞ this is [24,Theorem 12]). Also, if a co-echelon algebra of order 0 is contractible, then its completion is a Montel space [24,Theorem 13]. Now (iv) and (v) follow from Lemmas 5.8 and 5.9, respectively.
We conjecture that (v) holds for k ∞ (N 2 , V ) as well.