Contractibility of vector-valued Köthe echelon algebras

We give a characterization of those vector-valued Köthe echelon algebras which are contractible.


Introduction
Let S be some sequence (function) space, i.e. a topological vector space of sequences (functions) in which convergence in the original topology implies the coordinatewise (point-wise) convergence and let X be some topological vector space such that the space S(X) of X-valued sequences has a canonical definition. This is the case if e.g. S and X are Banach or Fréchet spaces. Assume that X satisfies some property P . It is a natural question to ask whether or not S(X) has the same property too. The specific property P that we are concerned with will be amenability/contractibility of an algebra. Such problems have already been considered in [4,14] where the authors study the relation between amenability properties of some Banach algebra A and the algebra C(Ω, A) of A-valued continuous functions on some compact Hausdorff space Ω.
In this paper, we will study the relation between contractibility of some Fréchet algebra A and a Köthe echelon algebra p (A) on the one hand and the A-valued Köthe echelon algebra p (A, A) on the other (see the next section for definitions). Recall that in the Banach algebra category contractibility seems to be a much too strong property. Indeed, if a contractible Banach algebra possesses the approximation property then it has to be finite dimensional-see [13,Theorem 4.1.2]. In the Fréchet algebra category the situation is different and there exist infinite dimensional contractible Fréchet algebras (even with a Schauder basis)-see [12,Theorem 14].
The next section collects all the necessary definitions and notation together with some well-known results while the last section contains the main result. For issues that are not explained here we refer the reader to [7,8] (functional analysis), [3] (topological algebra theory), [5] (homological methods).

Notation and preliminaries
We start by recalling some facts and definitions that will be used in the paper. It will be enough to restrict ourselves to the category of Fréchet spaces. Let A be a Fréchet algebra and let m be the multiplication in A . The product map is denoted by and it is the unique linearization of m, i.e. ∶ A � ⊗A → A and it is defined on elementary tensors as Let now P be a Fréchet A-bimodule. We say it is projective (see [3, We will also need an equivalent formulation. Throughout the article we denote by ℕ ∶= {1, 2, 3, …} the set of natural numbers and by the field of scalars which is either ℝ or ℂ . The sequence of standard unit vectors is denoted by (e j ) j∈ℕ . Let ( j ) j∈ℕ be a sequence of non-negative numbers. If there is some n ∈ ℕ such that j > 0 for all j ⩾ n then we say that j > 0 for large j ∈ ℕ.
A matrix A ∶= (a n (j)) n,j∈ℕ of non-negative numbers is a Köthe matrix if A Köthe matrix A ∶= (a n (j)) n,j∈ℕ is called bounded if a n ∈ ∞ for all n ∈ ℕ , i.e. if all the weights a n are bounded. For a Köthe matrix A the Köthe echelon space of order p, p (A) is the space of scalar-valued sequences defined as With the topology defined by the sequence of seminorms (‖ ⋅ ‖ n,p ) n∈ℕ they become Fréchet spaces. Clearly, 0 (A) is considered with the topology inherited from ∞ (A) . Equivalently-at least in the case when all the entries of the Köthe matrix are positive-one may say that p (A) is the intersection of Banach spaces p (a n ) (resp. c 0 (a n ) ) endowed with the weakest locally convex topology under which the natural injections p (A) ↪ p (a n ) are continuous for all n ∈ ℕ . A thorough investigation of Köthe echelon spaces can be found in [1] or [8,Chapter 27]. Observe that (e j ) j∈ℕ is a Schauder basis in p (A) if p is finite (this includes also p = 0).
In this paper we will be focused on those Köthe echelon spaces which are Fréchet algebras with respect to the coordinate-wise multiplication-see [2] for an expository account. A Köthe echelon space is a Fréchet algebra if and only if Without loss of generality we may assume that k ⩾ n , if necessary. Amenability/ contractibility of these algebras has been characterized by Pirkovskiĭ [9] and the author [10][11][12]. Below we recall those results.

) is nuclear and A is bounded.
(2) If p = 0, ∞ then the following conditions are equivalent: (3) If p = 0, ∞ then the following conditions are equivalent:

Main result
We are now in the position to state the main result of the paper.
‖x‖ n,p ∶= ‖(‖x j ‖ n a n (j)) j∈ℕ ‖ p (x ∈ p (A, X)). The proof splits into two separate cases depending on whether we have 1 ⩽ p < ∞ or p = 0 . Therefore the main theorem will be a simple consequence of the following two results. ⊗A are isomorphic as Fréchet algebras. Therefore as well. Let now be the right inverse bimodule maps to 1 and , respectively. The assignment gives rise to a continuous map ‖(u j ) j ‖ n,p = ‖u‖ n ‖ ‖ n,p < ∞ (n ∈ ℕ) ‖ ‖ n,p < ∞ (large n).
This is clearly a bimodule map since for elementary tensors we have Moreover,  0 (A, A) . Therefore for every n ∈ ℕ we have Since ‖u‖ n > 0 for large n this implies that ∈ 0 (A) . From Theorem 2.3 it now follows that 0 (A) is contractible.
(ii)⇒(i): Let u be the unit in A . Then u ⊗ = ∑ ∞ j=1 u ⊗ e j is the unit in 0 (A, A) and ‖u ⊗ ‖ n,∞ = ‖u‖ n ‖ ‖ n,∞ . It remains to show that there is a projective diagonal in 0 (A, A) � ⊗ 0 (A, A) . To this end, let the mapping ∶ A � ⊗ A → 0 (A, A) � ⊗ 0 (A, A) be defined as .
We need to show that is well-defined and continuous. Indeed, let a, b ∈ A and k ∈ ℕ . By the Rademacher averaging we get Thus for any > 0 and any element v = ∑ ∞ n=1 a n ⊗ b n satisfying Consequently, is well-defined and continuous. Let now be a projective diagonal. We will show that (d) is a projective diagonal in A) . To this end, let (a j ) j∈ℕ ∈ 0 (A, A) be given. Then (a n ⊗ e j ) ⊗ (b n ⊗ e j ).
(x n ⊗ e j ) ⊗ (y n a j ⊗ e j ).
We claim that these are equal. Indeed, let B ∈ ( 0 (A, )⊗ 0 (A, )) = ( 0 (A, ) × 0 (A, )) be a given continuous bilinear form. We define Clearly, each B j , j ∈ ℕ is continuous and bilinear. Since d ∈ A � ⊗ A is a projective diagonal, for any j ∈ ℕ we obtain We end this section with a remark on the case p = ∞ . From Theorem 2.3 it follows that if the algebra ∞ (A) is contractible then in particular, ∞ (A) = 0 (A) . Therefore we can mimic the proof of Proposition 3.3 to get that contractibility of ∞ (A) and A implies that of ∞ (A, A) . It is also clear that contractibility of ∞ (A, A) implies that of A . Therefore it is tempting to say that contractibility of ∞ (A) should follow from that of ∞ (A, A) . Since we do not have a proof of the last claim we state the following conjecture.

Conjecture 3.4 Let ∞ (A) be a Köthe echelon algebra and let
A be a Fréchet algebra. The following conditions are equivalent: B(a j x n ⊗ e j , y n ⊗ e j ) = ∞ ∑ n=1 B j (a j x n , y n ) B(x n ⊗ e j , y n a j ⊗ e j ). A)).
• (d) = ∞ ∑ j,n=1 x n y n ⊗ e j = ∞ ∑ j=1 u ⊗ e j Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/.