Contractibility of vector-valued Köthe echelon algebras

Abstract

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

1 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 coordinate-wise (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 ${\mathcal {P}}$. It is a natural question to ask whether or not S(X) has the same property too. The specific property ${\mathcal {P}}$ 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 $C(\Omega ,A)$ of A-valued continuous functions on some compact Hausdorff space $\Omega$.

In this paper, we will study the relation between contractibility of some Fréchet algebra ${{\mathcal {A}}}$ and a Köthe echelon algebra $\lambda _p(A)$ on the one hand and the ${{\mathcal {A}}}$-valued Köthe echelon algebra $\lambda _p(A,{{\mathcal {A}}})$ 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).

2 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 ${{\mathcal {A}}}$ be a Fréchet algebra and let m be the multiplication in ${{\mathcal {A}}}$. The product map is denoted by $\pi$ and it is the unique linearization of m, i.e. $\pi :{{\mathcal {A}}}{\widehat{\otimes }}{{\mathcal {A}}}\rightarrow {{\mathcal {A}}}$ and it is defined on elementary tensors as

$$\begin{aligned}\pi (a\otimes b):=ab.\end{aligned}$$

Let now P be a Fréchet A-bimodule. We say it is projective (see [3, Definition 2.8.35]) if for all Fréchet A-bimodules Y, Z, every admissible epimorphism $T\in {}_AL_A(Y,Z)$ and every $S\in {}_AL_A(P,Z)$ there exists $R\in {}_AL_A(P,Y)$ such that $T\circ R=S$, i.e. the diagram

is commutative. A Fréchet algebra A is biprojective if it is a projective A-bimodule. We will also need an equivalent formulation.

Proposition 2.1

[3, Proposition 2.8.41] A Fréchet algebra A is biprojective if and only if the product map is a retract, i.e. there exists a bimodule map $\sigma :A\rightarrow A{\widehat{\otimes }}A$ which is a right inverse to $\pi$.

The following result is crucial for our definition.

Theorem 2.2

[3, Theorem 2.8.48] Let A be a non-zero Fréchet algebra. Then the following conditions are equivalent:

• (i) every continuous derivation from A into any Fréchet A-bimodule is inner,

• (ii) A is unital and biprojective,

• (iii) A is unital and has a projective diagonal in $A{\widehat{\otimes }}A$, i.e. an element $d\in A{\widehat{\otimes }}A$ such that

  $$\begin{aligned}\pi (d)={\mathbf {1}}, \quad {}a\cdot d=d\cdot a{}(a\in A).\end{aligned}$$

A Fréchet algebra is contractible—see [6, Definition VII.1.59]—if it satisfies any of the conditions in Theorem 2.2.

Throughout the article we denote by ${{\mathbb {N}}}:=\{1,2,3,\ldots \}$ the set of natural numbers and by ${{\mathbb {K}}}$ the field of scalars which is either ${{\mathbb {R}}}$ or ${{\mathbb {C}}}$. The sequence of standard unit vectors is denoted by $(e_j)_{j\in {{\mathbb {N}}}}$. Let $(\xi _j)_{j\in {{\mathbb {N}}}}$ be a sequence of non-negative numbers. If there is some $n\in {{\mathbb {N}}}$ such that $\xi _j>0$ for all $j\geqslant n$ then we say that $\xi _j>0$ for large $j\in {{\mathbb {N}}}$.

A matrix $A:=(a_n(j))_{n,j\in {{\mathbb {N}}}}$ of non-negative numbers is a Köthe matrix if

• (i) $\forall \,j\in {{\mathbb {N}}}\,\,\exists \,n\in {{\mathbb {N}}}:\,\,\,\,a_n(j)>0,$

• (ii) $\forall \,j,n\in {{\mathbb {N}}}:\,\,\,\,a_n(j)\leqslant a_{n+1}(j)$.

A Köthe matrix $A:=(a_n(j))_{n,j\in {{\mathbb {N}}}}$ is called bounded if $a_n\in \ell _{\infty }$ for all $n\in {{\mathbb {N}}}$, i.e. if all the weights $a_n$ are bounded.

For a Köthe matrix A the Köthe echelon space of order p, $\lambda _p(A)$ is the space of scalar-valued sequences defined as

$$\begin{aligned}\lambda _p(A):=\big \{\xi \in {{\mathbb {K}}}^{{{\mathbb {N}}}}:\,\,\Vert \xi \Vert _{n,p}:=\Vert (\xi _ja_n(j))_{j\in {{\mathbb {N}}}}\Vert _{\ell _p}<\infty \,\,\text {for all}\,\,n\in {{\mathbb {N}}}\big \}\end{aligned}$$

if $1\leqslant p\leqslant \infty$ and

$$\begin{aligned}\lambda _0(A):=\big \{\xi \in {{\mathbb {K}}}^{{{\mathbb {N}}}}:\,\,\lim _{j\rightarrow \infty }\xi _ja_n(j)=0\,\,\text {for all}\,\,n\in {{\mathbb {N}}}\big \}.\end{aligned}$$

With the topology defined by the sequence of seminorms $(\Vert \cdot \Vert _{n,p})_{n\in {{\mathbb {N}}}}$ they become Fréchet spaces. Clearly, $\lambda _0(A)$ is considered with the topology inherited from $\lambda _{\infty }(A)$. Equivalently—at least in the case when all the entries of the Köthe matrix are positive—one may say that $\lambda _p(A)$ is the intersection of Banach spaces $\ell _p(a_n)$ (resp. $c_0(a_n)$) endowed with the weakest locally convex topology under which the natural injections $\lambda _p(A)\hookrightarrow \ell _p(a_n)$ are continuous for all $n\in {{\mathbb {N}}}$. A thorough investigation of Köthe echelon spaces can be found in [1] or [8, Chapter 27]. Observe that $(e_j)_{j\in {{\mathbb {N}}}}$ is a Schauder basis in $\lambda _p(A)$ 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

$$\begin{aligned} \forall \,n\in {{\mathbb {N}}}\,\,\exists \,k\in {{\mathbb {N}}}:\quad a_n/a_k^2\in \ell _{\infty }. \end{aligned}$$

(1)

Without loss of generality we may assume that $k\geqslant n$, if necessary. Amenability/contractibility of these algebras has been characterized by Pirkovskiĭ  [9] and the author [10,11,12]. Below we recall those results.

Theorem 2.3

[2, Theorems 8.2 and 8.4] Let $1\leqslant p\leqslant \infty$ or $p=0$ and let $\lambda _p(A)$ be a Köthe echelon algebra.

• (1) If $1\leqslant p<\infty$ then the following conditions are equivalent:

  • (i) $\lambda _p(A)$ is amenable,

  • (ii) $\lambda _p(A)$ is contractible,

  • (iii) $\lambda _p(A)$ is unital,

  • (iv) $\lambda _p(A)$ is nuclear and A is bounded.

• (2) If $p=0,\infty$ then the following conditions are equivalent:

  • (i) $\lambda _0(A)$ is contractible,

  • (ii) $\lambda _{\infty }(A)$ is contractible,

  • (iii) $\lambda _0(A)$ is unital.

• (3) If $p=0,\infty$ then the following conditions are equivalent:

  • (i) $\lambda _0(A)$ is amenable,

  • (ii) $\lambda _{\infty }(A)$ is amenable,

  • (iii) $\lambda _{\infty }(A)$ is unital,

  • (iv) A is bounded.

Let now $\lambda _p(A)$ be a Köthe echelon space and X some Fréchet space. The vector-valued Köthe echelon space $\lambda _p(A,X)$ is defined as

$$\begin{aligned}\lambda _p(A,X):=\{x=(x_j)_{j\in {{\mathbb {N}}}}\subset X:\,\,(\Vert x_j\Vert _na_n(j))_{j\in {{\mathbb {N}}}}\in \ell _p\,\,\,\,\forall \,n\in {{\mathbb {N}}}\}\end{aligned}$$

if $1\leqslant p\leqslant \infty$ and

$$\begin{aligned}\lambda _0(A,X):=\{x=(x_j)_{j\in {{\mathbb {N}}}}\subset X:\,\,(\Vert x_j\Vert _na_n(j))_{j\in {{\mathbb {N}}}}\in c_0\,\,\,\,\forall \,n\in {{\mathbb {N}}}\}.\end{aligned}$$

It comes equipped with the Fréchet space topology given by the sequence of semi-norms

$$\begin{aligned}\Vert x\Vert _{n,p}:=\Vert (\Vert x_j\Vert _na_n(j))_{j\in {{\mathbb {N}}}}\Vert _{\ell _p}\qquad (x\in \lambda _p(A,X)).\end{aligned}$$

Clearly, $\lambda _0(A,X)$ is a Fréchet subspace of $\lambda _{\infty }(A,X)$. If X and $\lambda _p(A)$ are Fréchet algebras then $\lambda _p(A,X)$ is a Fréchet algebra with coordinate-wise multiplication. The product map is clearly defined as $\pi _{\lambda _p(A,X)}:=\pi _{\lambda _p(A)}\otimes \pi _X$. For notational convenience we let $x\otimes e_j$ stand for the sequence in $\lambda _p(A,X)$ which has x on the j-th coordinate and zeros elsewhere. Observe that if p is finite (this includes $p=0$) then

$$\begin{aligned}x=\sum _{j=1}^{\infty }x_j\otimes e_j\qquad (x\in \lambda _p(A,X)).\end{aligned}$$

3 Main result

We are now in the position to state the main result of the paper.

Theorem 3.1

Let $1\leqslant p<\infty$ or $p=0$, let $\lambda _p(A)$ be a Köthe echelon algebra and let ${{\mathcal {A}}}$ be a Fréchet algebra. The vector-valued Köthe echelon algebra $\lambda _p(A,{{\mathcal {A}}})$ is contractible if and only if $\lambda _p(A)$ and ${{\mathcal {A}}}$ are both contractible.

The proof splits into two separate cases depending on whether we have $1\leqslant p<\infty$ or $p=0$. Therefore the main theorem will be a simple consequence of the following two results.

Proposition 3.2

Let $1\leqslant p<\infty$, let $\lambda _p(A)$ be Köthe echelon algebra and let ${{\mathcal {A}}}$ be a Fréchet algebra. Then $\lambda _p(A,{{\mathcal {A}}})$ is contractible if and only if $\lambda _p(A)$ and ${{\mathcal {A}}}$ are both contractible.

Proof

Necessity. The mapping

$$\begin{aligned} \theta :\lambda _p(A,{{\mathcal {A}}})\rightarrow {{\mathcal {A}}}.\qquad \theta ((x_j)_j):=x_1, \end{aligned}$$

is a surjective Fréchet algebra homomorphism therefore contractibility of ${{\mathcal {A}}}$ follows from [3, Proposition 2.8.64]. Let now $(u_j)_{j\in {{\mathbb {N}}}}$ be the unit in $\lambda _p(A,{{\mathcal {A}}})$. Then for any $k\in {{\mathbb {N}}}$ we have

$$\begin{aligned}(x\otimes e_k)(u_j)_j=xu_k\otimes e_k=x\otimes e_k=u_kx\otimes e_k=(u_j)_j(x\otimes e_k)\qquad (x\in \lambda _p(A,{{\mathcal {A}}})).\end{aligned}$$

Consequently, $u:=u_j=u_i$ for all $i,j\in {{\mathbb {N}}}$ and it is the unit in ${{\mathcal {A}}}$. This implies that $(u_j)_j=\sum _{j=1}^{\infty }u\otimes e_j$ and

$$\begin{aligned}\Vert (u_j)_j\Vert _{n,p}=\Vert u\Vert _n\Vert {\mathbf {1}}\Vert _{n,p}<\infty \qquad (n\in {{\mathbb {N}}})\end{aligned}$$

where we have denoted ${\mathbf {1}}:=\sum _{j=1}^{\infty }e_j$. Since $\Vert u\Vert _n>0$ for large n, we obtain

$$\begin{aligned}\Vert {\mathbf {1}}\Vert _{n,p}<\infty \qquad (\text {large}\,\,n).\end{aligned}$$

Hence $\lambda _p(A)$ is unital thus contractible by Theorem 2.3.

Sufficiency. From Theorem 2.3 if follows that $\lambda _p(A)$ is nuclear therefore [8, Proposition 28.16] implies that $\lambda _p(A)=\lambda _1(A)$ and, consequently, $\lambda _p(A,{{\mathcal {A}}})=\lambda _1(A,{{\mathcal {A}}})$. From [7, Corollary 15.7.2] it now follows that $\lambda _1(A,{{\mathcal {A}}})$ and $\lambda _1(A){\widehat{\otimes }}{{\mathcal {A}}}$ are isomorphic as Fréchet algebras. Therefore

$$\begin{aligned}\lambda _1(A){\widehat{\otimes }}\lambda _1(A){\widehat{\otimes }}{{\mathcal {A}}}{\widehat{\otimes }}{{\mathcal {A}}}=\lambda _1(A,{{\mathcal {A}}}){\widehat{\otimes }}\lambda _1(A,{{\mathcal {A}}})\end{aligned}$$

as well. Let now

$$\begin{aligned}\sigma _1:\lambda _1(A)\rightarrow \lambda _1(A){\widehat{\otimes }}\lambda _1(A),\qquad \sigma :{{\mathcal {A}}}\rightarrow {{\mathcal {A}}}{\widehat{\otimes }}{{\mathcal {A}}}\end{aligned}$$

be the right inverse bimodule maps to $\pi _1$ and $\pi$, respectively. The assignment

$$\begin{aligned}\xi \otimes x\mapsto \sigma _1(\xi )\otimes \sigma (x)\end{aligned}$$

gives rise to a continuous map

$$\begin{aligned}\sigma _1\otimes \sigma :\lambda _1(A){\widehat{\otimes }}{{\mathcal {A}}}\rightarrow \lambda _1(A){\widehat{\otimes }}\lambda _1(A){\widehat{\otimes }}{{\mathcal {A}}}{\widehat{\otimes }}{{\mathcal {A}}}.\end{aligned}$$

This is clearly a bimodule map since for elementary tensors we have

$$\begin{aligned} (\eta _1\otimes y_1)(\sigma _1\otimes \sigma (\xi \otimes x))(\eta _2\otimes y_2)&=\eta _1\sigma _1(\xi )\eta _2\otimes y_1\sigma (x)y_2 \\&=\sigma _1(\eta _1\xi \eta _2)\otimes \sigma (y_1xy_2) \\&=\sigma _1\otimes \sigma (\eta _1\xi \eta _2\otimes y_1xy_2) \\&=\sigma _1\otimes \sigma ((\eta _1\otimes y_1)(\xi \otimes x)(\eta _2\otimes y_2)). \end{aligned}$$

Moreover,

$$\begin{aligned}(\pi _1\otimes \pi )(\sigma _1\otimes \sigma (\xi \otimes x))=\pi _1\sigma _1(\xi )\otimes \pi \sigma (x)=\xi \otimes x.\end{aligned}$$

Thus $\sigma _1\otimes \sigma$ is a right inverse bimodule map and $\lambda _1(A,{{\mathcal {A}}})$ is biprojective.

By assumption $\lambda _1(A)$ and ${{\mathcal {A}}}$ are unital with units, say $\mathbf{1}$ and u, respectively. Then

$$\begin{aligned}\sum _{j=1}^{\infty }u\otimes e_j=u\otimes \mathbf{1}\end{aligned}$$

is the unit in $\lambda _p(A,{{\mathcal {A}}})$. Consequently, $\lambda _p(A,{{\mathcal {A}}})$ is contractible. $\square$

Proposition 3.3

Let $\lambda _0(A)$ be a Köthe echelon algebra and let ${{\mathcal {A}}}$ be a Fréchet algebra. The following conditions are equivalent:

• (i) $\lambda _0(A,{{\mathcal {A}}})$ is contractible,

• (ii) $\lambda _0(A)$ is contractible and ${{\mathcal {A}}}$ is contractible.

Proof

(i)$\Rightarrow$(ii): The mapping

$$\begin{aligned}\theta :\lambda _0(A,{{\mathcal {A}}})\rightarrow {{\mathcal {A}}}.\qquad \theta ((x_j)_j):=x_1,\end{aligned}$$

is a surjective Fréchet algebra homomorphism therefore contractibility of ${{\mathcal {A}}}$ follows from [3, Proposition 2.8.64]. Observe now that $u\otimes {\mathbf {1}}=\sum _{j=1}^{\infty }u\otimes e_j$ (u—the unit in ${{\mathcal {A}}}$) is the unit in $\lambda _0(A,{{\mathcal {A}}})$. Therefore for every $n\in {{\mathbb {N}}}$ we have

$$\begin{aligned}\Vert u\Vert _na_n(j)\xrightarrow [j\rightarrow \infty ]{}0.\end{aligned}$$

Since $\Vert u\Vert _n>0$ for large n this implies that ${\mathbf {1}}\in \lambda _0(A)$. From Theorem 2.3 it now follows that $\lambda _0(A)$ is contractible.

(ii)$\Rightarrow$(i): Let u be the unit in ${{\mathcal {A}}}$. Then $u\otimes {\mathbf {1}}=\sum _{j=1}^{\infty }u\otimes e_j$ is the unit in $\lambda _0(A,{{\mathcal {A}}})$ and $\Vert u\otimes {\mathbf {1}}\Vert _{n,\infty }=\Vert u\Vert _n\Vert {\mathbf {1}}\Vert _{n,\infty }$. It remains to show that there is a projective diagonal in $\lambda _0(A,{{\mathcal {A}}}){\widehat{\otimes }}\lambda _0(A,{{\mathcal {A}}})$. To this end, let the mapping $\iota :{{\mathcal {A}}}{{\,\mathrm{{\widehat{\otimes }}}\,}}{{\mathcal {A}}}\rightarrow \lambda _0(A,{{\mathcal {A}}}){{\,\mathrm{{\widehat{\otimes }}}\,}}\lambda _0(A,{{\mathcal {A}}})$ be defined as

$$\begin{aligned}\iota \Bigg (\sum _{n=1}^{\infty }a_n\otimes b_n\Bigg ):=\sum _{n,j=1}^{\infty }(a_n\otimes e_j)\otimes (b_n\otimes e_j).\end{aligned}$$

We need to show that $\iota$ is well-defined and continuous. Indeed, let $a,b\in {{\mathcal {A}}}$ and $k\in {{\mathbb {N}}}$. By the Rademacher averaging we get

$$\begin{aligned} \Vert \iota (a\otimes b)\Vert _{k,\infty }&=\Bigg \Vert \sum _{j=1}^{\infty }(a\otimes e_j)\otimes (b\otimes e_j)\Bigg \Vert _{k,\infty } \\&=\Bigg \Vert \int _0^1\Bigg (\sum _{j=1}^{\infty }r_j(t)a\otimes e_j\Bigg )\otimes \Bigg (\sum _{j=1}^{\infty }r_j(t)b\otimes e_j\Bigg )\mathrm{d}t\Bigg \Vert _{k,\infty } \\&\leqslant \sup _{t\in [0,1]}\Bigg \Vert \Bigg (\sum _{j=1}^{\infty }r_j(t)a\otimes e_j\Bigg )\Vert _{k,\infty }\Bigg \Vert \Bigg (\sum _{j=1}^{\infty }r_j(t)b\otimes e_j\Bigg )\Bigg \Vert _{k,\infty } \\&\leqslant \Bigg \Vert \Bigg (\sum _{j=1}^{\infty }a\otimes e_j\Bigg )\Vert _{k,\infty }\Bigg \Vert \Bigg (\sum _{j=1}^{\infty }b\otimes e_j\Bigg )\Vert _{k,\infty } \\&=\Vert {\mathbf {1}}\Vert _{k,\infty }^2\Vert a\Vert _k\Vert b\Vert _k. \end{aligned}$$

Thus for any $\varepsilon >0$ and any element $v=\sum _{n=1}^{\infty }a_n\otimes b_n$ satisfying

$\sum _{n=1}^{\infty }\Vert a_n\Vert _k\Vert b_n\Vert _k<\Vert v\Vert _{{{\mathcal {A}}}{{\,\mathrm{{\widehat{\otimes }}}\,}}{{\mathcal {A}}},k}+\varepsilon$ we obtain

$$\begin{aligned}\Vert \iota (v)\Vert _k\leqslant \Vert {\mathbf {1}}\Vert _{k,\infty }^2\sum _{n=1}^{\infty }\Vert a_n\Vert _k\Vert b_n\Vert _k<\Vert {\mathbf {1}}\Vert _{k,\infty }^2(\Vert v\Vert _{{{\mathcal {A}}}{{\,\mathrm{{\widehat{\otimes }}}\,}}{{\mathcal {A}}},k}+\varepsilon ).\end{aligned}$$

Hence

$$\begin{aligned}\Vert \iota (v)\Vert _k\leqslant \Vert {\mathbf {1}}\Vert _{k,\infty }^2\Vert v\Vert _{{{\mathcal {A}}}{{\,\mathrm{{\widehat{\otimes }}}\,}}{{\mathcal {A}}},k}\qquad (v\in {{\mathcal {A}}}{{\,\mathrm{{\widehat{\otimes }}}\,}}{{\mathcal {A}}}).\end{aligned}$$

Consequently, $\iota$ is well-defined and continuous. Let now

$$\begin{aligned}d:=\sum _{n=1}^{\infty }x_n\otimes y_n\in {{\mathcal {A}}}{\widehat{\otimes }}{{\mathcal {A}}}\end{aligned}$$

be a projective diagonal. We will show that $\iota (d)$ is a projective diagonal in $\lambda _0(A,{{\mathcal {A}}}){\widehat{\otimes }}\lambda _0(A,{{\mathcal {A}}})$. To this end, let $(a_j)_{j\in {{\mathbb {N}}}}\in \lambda _0(A,{{\mathcal {A}}})$ be given. Then

$$\begin{aligned}(a_j)_j\cdot \iota (d)=\sum _{n,j=1}^{\infty }(a_jx_n\otimes e_j)\otimes (y_n\otimes e_j)\end{aligned}$$

and

$$\begin{aligned}\iota (d)\cdot (a_j)_j=\sum _{n,j=1}^{\infty }(x_n\otimes e_j)\otimes (y_na_j\otimes e_j).\end{aligned}$$

We claim that these are equal. Indeed, let $B\in (\lambda _0(A,{{\mathcal {A}}}){{\,\mathrm{{\widehat{\otimes }}}\,}}\lambda _0(A,{{\mathcal {A}}}))'={{\mathcal {B}}}(\lambda _0(A,{{\mathcal {A}}})\times \lambda _0(A,{{\mathcal {A}}}))$ be a given continuous bilinear form. We define

$$\begin{aligned}B_j:{{\mathcal {A}}}\times {{\mathcal {A}}}\rightarrow {{\mathbb {K}}},\quad B_j(a,b):=B(a\otimes e_j,b\otimes e_j)\qquad (j\in {{\mathbb {N}}}).\end{aligned}$$

Clearly, each $B_j,\,j\in {{\mathbb {N}}}$ is continuous and bilinear. Since $d\in {{\mathcal {A}}}{{\,\mathrm{{\widehat{\otimes }}}\,}}{{\mathcal {A}}}$ is a projective diagonal, for any $j\in {{\mathbb {N}}}$ we obtain

$$\begin{aligned} \sum _{n=1}^{\infty }B(a_jx_n\otimes e_j,y_n\otimes e_j)&=\sum _{n=1}^{\infty }B_j(a_jx_n,y_n) \\&=B_j(a_j\cdot d)=B_j(d\cdot a_j) \\&=\sum _{n=1}^{\infty }B(x_n\otimes e_j,y_na_j\otimes e_j). \end{aligned}$$

Consequently,

$$\begin{aligned}B\big ((a_j)_j\cdot \iota (d)\big )=B\big (\iota (d)\cdot (a_j)_j\big )\end{aligned}$$

for every continuous bilinear form B. By Hahn–Banach we finally get

$$\begin{aligned}(a_j)_j\cdot \iota (d)=\iota (d)\cdot (a_j)_j\qquad ((a_j)_j\in \lambda _0(A,{{\mathcal {A}}})).\end{aligned}$$

Moreover,

$$\begin{aligned}\pi \circ \iota (d)=\sum _{j,n=1}^{\infty }x_ny_n\otimes e_j=\sum _{j=1}^{\infty }u\otimes e_j\end{aligned}$$

is the unit in $\lambda _0(A,{{\mathcal {A}}})$. Therefore $\iota (d)$ is a projective diagonal in the space $\lambda _0(A,{{\mathcal {A}}}){{\,\mathrm{{\widehat{\otimes }}}\,}}\lambda _0(A,{{\mathcal {A}}})$. From Theorem 2.2 it now follows that $\lambda _{\infty }(A,{{\mathcal {A}}})=\lambda _0(A,{{\mathcal {A}}})$ is contractible and the proof is thereby complete. $\square$

We end this section with a remark on the case $p=\infty$. From Theorem 2.3 it follows that if the algebra $\lambda _{\infty }(A)$ is contractible then in particular, $\lambda _{\infty }(A)=\lambda _0(A)$. Therefore we can mimic the proof of Proposition 3.3 to get that contractibility of $\lambda _{\infty }(A)$ and ${{\mathcal {A}}}$ implies that of $\lambda _{\infty }(A,{{\mathcal {A}}})$. It is also clear that contractibility of $\lambda _{\infty }(A,{{\mathcal {A}}})$ implies that of ${{\mathcal {A}}}$. Therefore it is tempting to say that contractibility of $\lambda _{\infty }(A)$ should follow from that of $\lambda _{\infty }(A,{{\mathcal {A}}})$. Since we do not have a proof of the last claim we state the following conjecture.

Conjecture 3.4

Let $\lambda _{\infty }(A)$ be a Köthe echelon algebra and let ${{\mathcal {A}}}$ be a Fréchet algebra. The following conditions are equivalent:

• (i) $\lambda _{\infty }(A,{{\mathcal {A}}})$ is contractible,

• (ii) $\lambda _{\infty }(A)$ is contractible and ${{\mathcal {A}}}$ is contractible.