On Orlicz sequence algebras

Abstract

We study Orlicz sequence algebras and their properties. In particular, we fully characterize biflat and biprojective Orlicz sequence algebras as well as weakly amenable and approximately (semi-)amenable Orlicz sequence algebras. As a consequence, we show the existence of a wide class of sequence algebras that behave differently—in terms of the amenability properties—from any of the algebras $\ell _{p}, 1\leqslant p\leqslant \infty$.

1 Introduction

The aim of this note is to study Orlicz sequence spaces $\ell _{\varphi }$ and $h_{\varphi }$ as Banach algebras. Recall that an Orlicz function space $L^{\varphi }(X)$ is a Banach algebra with respect to pointwise multiplication if and only if $b_{\varphi }<\infty$ (the definition of the number $b_{\varphi }$ is provided at the beginning of Sect. 2) or X is at most a countable union of atoms—see [7, Theorem B]. Consequently, Orlicz sequence spaces are always Banach algebras with respect to pointwise multiplication. On the other hand, if G is a locally compact group then $L^{\varphi }(G)$ is a Banach algebra if and only if $L^{\varphi }(G)$ embeds continuously into $L^{1}(G)$—see [8, Theorem 2]. The situation changes substantially—see, e.g., [22, 25]—if one considers weighted Orlicz spaces $L^{\varphi }_{w}(G)$, where by a weight we mean a measurable function $w\colon G\to (0,\infty )$ satisfying

$$\begin{aligned} w(st)\leq w(s)w(t)\quad (s,t\in G). \end{aligned}$$

One can also consider the so-called twisted Orlicz algebras, which are Orlicz spaces $L^{\varphi }(G)$ with multiplication arising from a specific 2-cocycle—see [23, 24] for details. Their weighted variants are also considered. In this note, we focus entirely on Orlicz sequence algebras $\ell _{\varphi }$ and $h_{\varphi }$.

The paper is organized as follows. Section 2 gives some preliminaries and notation. Section 3 contains a list of auxiliary but very useful results. The final section presents the main results of the paper. It emphasizes in particular the power of the $\Delta _{2}(0)$ condition.

For unexplained details from Banach algebra theory we refer the reader to [3, 27] and from Orlicz space theory to [12–16, 18, 20, 26].

2 Notation and preliminaries

In the whole paper φ denotes an Orlicz function (see [12, 16, 18, 20, 26]), that is, $\varphi \colon [0,\infty )\rightarrow [0,\infty ]$ and φ is convex, vanishing, and right continuous at zero, not identically equal to zero and left continuous on the interval $(0,\infty )$. By the convexity of φ, it is nondecreasing on $[0,\infty )$. Let

$$\begin{aligned} &a_{\varphi }:=\sup \bigl\lbrace u\geq 0: \varphi (u )=0 \bigr\rbrace , \\ &b_{\varphi }:=\sup \bigl\lbrace u\geq 0: \varphi (u )< \infty \bigr\rbrace . \end{aligned}$$

Note that the left continuity of φ on $(0, \infty )$ is equivalent to the fact that

$$\begin{aligned} \lim_{u\to (b_{\varphi })^{-}}\varphi (u )=\varphi (b_{\varphi } ). \end{aligned}$$

Recall that an Orlicz function φ is an N-function at 0 if $a_{\varphi }=0$ and $\lim_{u\rightarrow 0}\frac{\varphi (u)}{u}=0$.

The generalized inverse $\varphi ^{-1}$ of the Orlicz function φ is defined as follows:

$$\begin{aligned} \varphi ^{-1}(v):=\inf \bigl\{ u\geq 0\colon \varphi (u)>v\bigr\} \end{aligned}$$

if $v\in [0,\infty )$ and

$$\begin{aligned} \varphi ^{-1}(\infty )=\lim_{v\rightarrow \infty}\varphi ^{-1}(v) \end{aligned}$$

(see [21] and [11]).

We say that an Orlicz function φ satisfies the condition $\Delta _{2}(0)$ (in short, $\varphi \in \Delta _{2}(0)$) if there exist $u_{0}>0$ and a constant $K>0$ such that $\varphi (u_{0})>0$ and $\varphi (2u)\leq K\varphi (u)$ for every $u\leq u_{0}$ (then we also have $a_{\varphi }=0$).

Given any Orlicz function φ, we define its complementary function in the sense of Young by the formula

$$\begin{aligned} \psi (u):=\sup_{v>0}\bigl\{ uv-\varphi (v)\bigr\} \end{aligned}$$

(1)

for all $u\geq 0$. It is easy to show that ψ is also an Orlicz function and φ is complementary to ψ (see [16, p. 147]).

For any Orlicz function φ we define Orlicz sequence spaces $\ell _{\varphi }$ by

$$\begin{aligned} \ell _{\varphi }:= \Biggl\{ x=\bigl(x(i)\bigr)_{i\in {\mathbb{N}}}\in { \mathbb{K}}^{{ \mathbb{N}}}\colon \sum_{i=1}^{\infty} \varphi \bigl(\lambda \bigl\vert x(i) \bigr\vert \bigr)< \infty \text{ for some }\lambda >0 \Biggr\} , \end{aligned}$$

where ${\mathbb{K}}$ is either ${\mathbb{R}}$ or ${\mathbb{C}}$. It is well known that the space $\ell _{\varphi }$ equipped with the Luxemburg norm $\Vert \cdot \Vert _{\varphi }$, defined by

$$\begin{aligned} \Vert x \Vert _{\varphi }:=\inf \Biggl\{ \lambda >0\colon \sum _{i=1}^{\infty} \varphi \bigl(\lambda ^{-1} \bigl\vert x(i) \bigr\vert \bigr)\leq 1 \Biggr\} , \end{aligned}$$

is a Banach space. Moreover, as has been shown in [7], it is also a Banach algebra.

Recall that the subspace $h_{\varphi }$ of the space $\ell _{\varphi }$ is defined by the formula

$$\begin{aligned} h_{\varphi }:= \Biggl\{ x\in \ell _{\varphi } | \ \forall \lambda >0 \ \exists i_{0}\in {\mathbb{N}}\colon \sum_{i=i_{0}}^{\infty} \varphi \bigl(\lambda \bigl\vert x(i) \bigr\vert \bigr)< \infty \Biggr\} . \end{aligned}$$

Let $E\subset {\mathbb{K}}^{{\mathbb{N}}}$ be a Banach lattice. An element $x\in E$ is said to be order continuous if for any sequence $(x_{n} )$ in E with $0\leq x_{n}\leq \vert x\vert$ and $x_{n}\rightarrow 0$ coordinatewise one obtains $\Vert x_{n}\Vert _{E}\rightarrow 0$. The subspace $E_{a}$ of all order-continuous elements in E is an order ideal in E. The space E is called order continuous if $E_{a}=E$, see [17]. It is well known that $h_{\varphi }=(\ell _{\varphi })_{a}$. Hence, the standard unit vectors $e_{n}:=(\delta _{jn})_{j\in {\mathbb{N}}}, n\in {\mathbb{N}}$ constitute a Schauder basis in $h_{\varphi }$ (see also [16, Proposition 4.a.2]). The sequence $(e_{n}^{*})_{n\in {\mathbb{N}}}$ of evaluation functionals is defined as

$$\begin{aligned} \bigl\langle x,e_{n}^{*}\bigr\rangle :=x(n)\quad (x\in \ell _{\varphi }, n\in {\mathbb{N}}). \end{aligned}$$

It is easy to show that if $a_{\varphi }>0$, then $\ell _{\varphi }=\ell _{\infty}$ ($h_{\varphi }=c_{0}$) as sets and the norms $\|\cdot \|_{\varphi }$ and $\|\cdot \|_{\infty}$ are equivalent. On the other hand, if $a_{\varphi }=0$, then $\ell _{\varphi }\hookrightarrow c_{0}$. Moreover, in this case $\ell _{\varphi }=h_{\varphi }$ if and only if $\varphi \in \Delta _{2}(0)$ (see [16, Proposition 4.a.4]). Recall that if φ vanishes only at zero but is not an N-function at 0, then $\ell _{\varphi }=\ell _{1}$ as sets and the norms $\|\cdot \|_{\varphi }$ and $\|\cdot \|_{1}$ are equivalent (see [16, p. 147]). On the other hand, there exist N-functions at 0 such that $\ell _{\varphi }$ is not isomorphic to any $\ell _{p}$, $1\leq p\leq \infty$. Obviously, we have this situation whenever $\varphi \notin \Delta _{2}(0)$, because then $h_{\varphi }$ is a proper subspace of $\ell _{\varphi }$. A more sophisticated construction is provided in [14, Theorem 3].

We end this section by recalling a number of well-known ideas. If X is a Banach space then by $\iota \colon X\hookrightarrow X''$ we denote the canonical embedding into the second dual. If A is a Banach algebra then the product map $\pi _{A}\colon A\widehat{\otimes}A\to A$ is the unique linear mapping arising from the bilinear map $A\times A\ni (a,b)\mapsto ab\in A$. We will simply write π when there is no risk of confusion. If, furthermore, X is a Banach A-bimodule then $X'$ becomes the so-called dual A-bimodule with the bimodule operations defined as

$$\begin{aligned} \langle x,a\cdot \lambda \rangle:=\langle x\cdot a,\lambda \rangle \quad\text{and}\quad \langle x,\lambda \cdot a\rangle:=\langle a \cdot x,\lambda \rangle\quad \bigl(a\in A, x\in X, \lambda \in X'\bigr). \end{aligned}$$

(2)

A projective tensor product $X\widehat {\otimes }X$ becomes canonically an A-bimodule with the bimodule operations defined as

$$\begin{aligned} a\cdot (x\otimes y):=a\cdot x\otimes y,\qquad (x\otimes y)\cdot a:=x \otimes y\cdot a\quad (a\in A, x,y\in X). \end{aligned}$$

A bounded linear map $T\colon X\to Y$ between A-bimodules X and Y is a bimodule map if

$$\begin{aligned} T(a\cdot x\cdot b)=a\cdot Tx\cdot b \quad (a,b\in A, x\in X). \end{aligned}$$

If A is a Banach algebra and X is a Banach A-bimodule then a linear (not necessarily bounded) mapping $\delta \colon A\to X$ is called a derivation if it satisfies the so-called “derivation rule”, i.e.,

$$\begin{aligned} \delta (ab)=a\cdot \delta (b)+\delta (a)\cdot b \quad(a,b\in A). \end{aligned}$$

A derivation $\delta \colon A\to X$ is inner if there is some $x\in X$ such that

$$\begin{aligned} \delta (a)=\operatorname {ad}_{x}(a):=a\cdot x-x\cdot a\quad (a\in A). \end{aligned}$$

A Banach algebra A is said to be amenable if for any Banach A-bimodule X every continuous derivation $\delta \colon A\to X'$ into the dual A-bimodule $X'$ (see (2)) is inner.

3 Auxiliary results

Recall that if A is a Banach algebra then we denote $A^{2}:=\operatorname{span}\{ab\colon a,b\in A\}$ and $\overline{A^{2}}$ is the norm closure of $A^{2}$.

Proposition 3.1

If φ is an Orlicz function and $a_{\varphi }=0$, then $\overline{(\ell _{\varphi })^{2}}=h_{\varphi }$.

Proof

Clearly, $e_{n}=e_{n}\cdot e_{n}\in (\ell _{\varphi })^{2}$, thus $\sum_{i=1}^{j}x(i)e_{i}\in (\ell _{\varphi })^{2}$ for any $x\in h_{\varphi }$ and any $j\in {\mathbb{N}}$ and, in consequence, $h_{\varphi }\subset \overline{(\ell _{\varphi })^{2}}$.

Conversely, let $x,y\in \ell _{\varphi }$ and let $\mu >0$ satisfy $\|y\|_{\varphi }<\mu$. Since $\ell _{\varphi }\subset c_{0}$, for an arbitrary $\lambda >0$ there exists $i_{0}\in {\mathbb{N}}$ such that

$$\begin{aligned} \lambda \bigl\vert x(i) \bigr\vert \leq \mu ^{-1}\quad\text{for any }i \geq i_{0}. \end{aligned}$$

Hence,

$$\begin{aligned} \sum_{i=i_{0}}^{\infty}\varphi \bigl(\lambda \bigl\vert x(i)y(i) \bigr\vert \bigr)\leq \sum_{i=i_{0}}^{ \infty} \varphi \bigl(\mu ^{-1} \bigl\vert y(i) \bigr\vert \bigr)\leq 1< \infty. \end{aligned}$$

Consequently, $\overline{(\ell _{\varphi })^{2}}\subset h_{\varphi }$. □

Theorem 3.2

If φ is an Orlicz function and $a_{\varphi }=0$ then the Orlicz sequence algebras $\ell _{\varphi }$ and $h_{\varphi }$ are not closed with respect to taking square roots.

Proof

Let $u_{1}\leq 10^{-2}$ satisfy $\varphi (u_{1})\leq \frac{1}{2}$ and let $k_{1}\in {\mathbb{N}}$ be the largest number such that

$$\begin{aligned} \frac{1}{4}< k_{1}\varphi (u_{1})\leq \frac{1}{2}. \end{aligned}$$

Let $u_{2}\leq \min \{u_{1},10^{-4}\}$ satisfy $\varphi (2u_{2})\leq \frac{1}{4}$ and let $k_{2}\in {\mathbb{N}}$ be the largest number such that

$$\begin{aligned} \frac{1}{8}< k_{2}\varphi (2u_{2})\leq \frac{1}{4}. \end{aligned}$$

Proceeding recursively, we obtain two sequences $(u_{n})_{n\in {\mathbb{N}}}\subset (0,\infty )$ and $(k_{n})_{n\in {\mathbb{N}}}\subset {\mathbb{N}}$ such that

$$\begin{aligned} u_{n}\leq \min \bigl\{ u_{n-1},10^{-2n}\bigr\} ,\qquad \frac{1}{2^{n+1}}< k_{n} \varphi (nu_{n})\leq \frac{1}{2^{n}} \quad (n\in {\mathbb{N}}). \end{aligned}$$

Let $l_{0}=0$, $l_{n}=\sum_{m=1}^{n}k_{m}$ and

$$\begin{aligned} x:=\sum_{n=1}^{\infty}\sum _{i=l_{n-1}+1}^{l_{n}}u_{n}e_{i}. \end{aligned}$$

We will show that $x\in h_{\varphi }$ and $\sqrt{x}\notin \ell _{\varphi }$. To this end, let $\lambda >0$ be fixed and let $n_{0}$ be the smallest natural number such that $\lambda \leq n_{0}$. Then,

$$\begin{aligned} \sum_{i=l_{n_{0}-1}+1}^{\infty}\varphi \bigl(\lambda x(i) \bigr) & \leq \sum_{n=n_{0}}^{ \infty}\sum _{i=l_{n-1}+1}^{l_{n}}\varphi (nu_{n}) \\ & \leq \sum_{n=n_{0}}^{\infty}k_{n} \varphi (nu_{n})\leq \sum_{n=n_{0}}^{ \infty} \frac{1}{2^{n}}\leq 1< \infty. \end{aligned}$$

Consequently, $x\in h_{\varphi }$. To obtain the other claim we again fix $\lambda >0$ and choose $n_{0}\in {\mathbb{N}}$ such that $\lambda >\frac{n}{5^{n}}$ for every $n\geq n_{0}$. Then,

$$\begin{aligned} \varphi (\lambda \sqrt{u_{n}})\geq \varphi \biggl(\frac{n}{5^{n}} \frac{u_{n}}{\sqrt{u_{n}}} \biggr)\geq \varphi \bigl(2^{n}nu_{n} \bigr)\geq 2^{n} \varphi (nu_{n}), \end{aligned}$$

where the last inequality follows from convexity of φ. Thus,

$$\begin{aligned} \sum_{i=1}^{\infty}\varphi \bigl(\lambda \sqrt{x(i)}\bigr)=\sum_{n=1}^{\infty} \sum _{i=l_{n-1}+1}^{l_{n}}\varphi (\lambda \sqrt{u_{n}})\geq \sum_{n=n_{0}}^{ \infty}2^{n}k_{n} \varphi (nu_{n})\geq \sum_{n=n_{0}}^{\infty} \frac{1}{2}=\infty. \end{aligned}$$

Consequently, $\sqrt{x}\notin \ell _{\varphi }$. □

Remark 3.3

From [27, Proposition 2.2.1] and [3, Corollary 2.9.25] it follows that if $a_{\varphi }=0$ then neither $\ell _{\varphi }$ nor $h_{\varphi }$ is amenable.

Recall that a Banach algebra A factors weakly if $A^{2}=A$ as sets.

Corollary 3.4

If φ is an Orlicz function and $a_{\varphi }=0$, then the Orlicz sequence algebras $\ell _{\varphi }$ and $h_{\varphi }$ do not factor weakly.

Proof

Let $x\in h_{\varphi }$ be such that $\sqrt{x}\notin \ell _{\varphi }$. Assume towards a contradiction that $x\in (\ell _{\varphi })^{2}$, i.e.,

$$\begin{aligned} x=\sum_{k=1}^{n}a_{k}b_{k}\quad (a_{k},b_{k}\in \ell _{\varphi }). \end{aligned}$$

Then,

$$\begin{aligned} \forall i\in {\mathbb{N}} \ \exists k=1,\ldots,n\colon \quad\bigl\vert a_{k}(i)b_{k}(i) \bigr\vert \geq \frac{1}{n} \bigl\vert x(i) \bigr\vert . \end{aligned}$$

We can therefore decompose ${\mathbb{N}}$ into a finite disjoint union ${\mathbb{N}}=\bigcup_{k=1}^{n}N_{k}$ and

$$\begin{aligned} \forall k=1,\ldots,n \ \forall i\in N_{k}\colon\quad \bigl\vert a_{k}(i)b_{k}(i) \bigr\vert \geq \frac{1}{n} \bigl\vert x(i) \bigr\vert . \end{aligned}$$

This implies

$$\begin{aligned} \forall k=1,\ldots,n \ \forall i\in N_{k}\colon\quad \bigl\vert x(i) \bigr\vert ^{ \frac{1}{2}}\leq \sqrt{n}\max \bigl\{ \bigl\vert a_{k}(i) \bigr\vert , \bigl\vert b_{k}(i) \bigr\vert \bigr\} \leq \sqrt{n} \bigl( \bigl\vert a_{k}(i) \bigr\vert + \bigl\vert b_{k}(i) \bigr\vert \bigr). \end{aligned}$$

Let

$$\begin{aligned} \lambda:=2\sqrt{n}\max \bigl\{ \Vert a_{k} \Vert _{\varphi }, \Vert b_{k} \Vert _{\varphi } \colon k=1,\ldots,n\bigr\} . \end{aligned}$$

Using the convexity of φ we obtain

$$\begin{aligned} \sum_{i=1}^{\infty}\varphi \bigl(\lambda ^{-1} \bigl\vert x(i) \bigr\vert ^{1/2}\bigr) & =\sum _{k=1}^{n} \sum _{i\in N_{k}}\varphi \bigl(\lambda ^{-1} \bigl\vert x(i) \bigr\vert ^{1/2}\bigr) \\ & \leq \sum_{k=1}^{n}\sum _{i\in N_{k}}\varphi \biggl( \frac{ \vert a_{k}(i) \vert }{2 \Vert a_{k} \Vert _{\varphi }}+ \frac{ \vert b_{k}(i) \vert }{2 \Vert b_{k} \Vert _{\varphi }} \biggr) \\ & \leq \frac{1}{2}\sum_{k=1}^{n}\sum _{i\in N_{k}} \bigl(\varphi \bigl( \Vert a_{k} \Vert _{\varphi }^{-1} \bigl\vert a_{k}(i) \bigr\vert \bigr)+\varphi \bigl( \Vert b_{k} \Vert _{\varphi }^{-1} \bigl\vert b_{k}(i) \bigr\vert \bigr) \bigr) \\ & \leq \frac{1}{2}\sum_{k=1}^{n}\sum _{i=1}^{\infty} \bigl(\varphi \bigl( \Vert a_{k} \Vert _{\varphi }^{-1} \bigl\vert a_{k}(i) \bigr\vert \bigr)+\varphi \bigl( \Vert b_{k} \Vert _{\varphi }^{-1} \bigl\vert b_{k}(i) \bigr\vert \bigr) \bigr)\leq n< \infty. \end{aligned}$$

Consequently, $\sqrt{x}\in \ell _{\varphi }$, which contradicts the choice of x. □

We are now going to focus on Arens regularity. Recall that if A is a Banach algebra then $A''$ can be made into a Banach algebra in two canonical ways. These are the so-called Arens products and they are defined as follows. Let $\Phi,\Psi \in A''$ be given. Then, the first Arens product is defined as

where

$$\begin{aligned} \langle a,\Psi \cdot \lambda \rangle:=\langle \lambda \cdot a,\Psi \rangle \quad(a\in A) \end{aligned}$$

and

$$\begin{aligned} \langle b,\lambda \cdot a\rangle:=\langle ab,\lambda \rangle\quad (b \in A). \end{aligned}$$

The second Arens product is defined as

$$\begin{aligned} \langle \lambda,\Phi \diamond \Psi \rangle:=\langle \lambda \cdot \Phi,\Psi \rangle \quad\bigl(\lambda \in A'\bigr), \end{aligned}$$

where

$$\begin{aligned} \langle a,\lambda \cdot \Phi \rangle:=\langle a\cdot \lambda,\Phi \rangle\quad (a\in A) \end{aligned}$$

and

$$\begin{aligned} \langle b,a\cdot \lambda \rangle:=\langle ba,\lambda \rangle\quad (b \in A). \end{aligned}$$

A Banach algebra A is called Arens regular if for all $\Phi,\Psi \in A''$.

Proposition 3.5

Orlicz sequence algebras $\ell _{\varphi }$ and $h_{\varphi }$ are Arens regular.

Proof

If $a_{\varphi }>0$ then $\ell _{\varphi }=\ell _{\infty}$ (equivalent norms) and $\ell _{\infty}$, being a unital $C^{*}$-algebra, is Arens regular by [2, Theorem 7.1] (cf. [3, Theorem 3.2.36]). From now on we restrict ourselves to the case where $a_{\varphi }=0$. Assume for a moment that $a\in \ell _{\varphi }$ and $\theta \in \ell _{\varphi }'$ are given so that $a\cdot \theta \in \ell _{\varphi }'$. From [26, Proposition 2] it follows that

$$\begin{aligned} \ell _{\varphi }'=\ell _{\psi}\oplus h_{\varphi }^{\perp}, \end{aligned}$$

where ψ denotes the complementary function to φ (see formula (1)). Therefore, there exists a unique decomposition

$$\begin{aligned} \theta =(y,f),\quad y\in \ell _{\psi}, f\in h_{\varphi }^{\perp}. \end{aligned}$$

Thus,

$$\begin{aligned} \langle x,a\cdot \theta \rangle =\langle xa,y\rangle +\langle xa,f \rangle\quad (x\in \ell _{\varphi }). \end{aligned}$$

From Proposition 3.1 it now follows that $xa\in h_{\varphi }$, whence

$$\begin{aligned} \langle xa,f\rangle =0 \end{aligned}$$

and, consequently,

$$\begin{aligned} \langle x,a\cdot \theta \rangle =\langle x,ay\rangle \quad (x\in \ell _{\varphi }). \end{aligned}$$

Since $ay\in h_{\psi}$ (again the argument from the proof of Proposition 3.1 applies), we may assume without loss of generality that

$$\begin{aligned} \forall a\in \ell _{\varphi }, \theta \in \ell _{\varphi }' \colon\quad a\cdot \theta \in \iota (h_{\psi}). \end{aligned}$$

We now proceed with the proof of Arens regularity. Since $h_{\psi}'=\ell _{\varphi }$ (see [16, Proposition 4.b.1], recall that φ is the complementary function of ψ) we may apply the Dixmier projection to obtain the decomposition

$$\begin{aligned} \ell _{\varphi }''=\ell _{\varphi }\oplus \iota (h_{\psi})^{\perp}. \end{aligned}$$

Let $(a,F),(b,G)\in \ell _{\varphi }''$ with $a,b\in \ell _{\varphi },F,G\in \iota (h_{\psi})^{\perp}$ be given. For any $\theta \in \ell _{\varphi }'$ we obtain

By the choice of the elements involved we obtain

$$\begin{aligned} \langle a,G\cdot \theta \rangle =\langle \theta \cdot a,G\rangle =0, \qquad\langle b \cdot \theta,F\rangle =0, \qquad\langle G\cdot \theta,F \rangle =0. \end{aligned}$$

Therefore,

whence

Similarly, we show that

$$\begin{aligned} (a,F)\diamond (b,G)=(ab,0). \end{aligned}$$

 □

Recall that a character space $\Phi _{A}$ of a Banach algebra A is a set of nonzero multiplicative functionals on A.

Proposition 3.6

Let φ be an Orlicz function and $a_{\varphi }=0$. Then,

$$\begin{aligned} \Phi _{\ell _{\varphi }}=\Phi _{h_{\varphi }}=\bigl\{ e_{n}^{*} \colon n \in {\mathbb{N}}\bigr\} . \end{aligned}$$

Proof

Let $f\in \ell _{\varphi }'$ be a multiplicative functional. We have

$$\begin{aligned} f(e_{n})=f\bigl(e_{n}^{2}\bigr)=f(e_{n})^{2}\quad (n\in {\mathbb{N}}) \end{aligned}$$

therefore,

$$\begin{aligned} f(e_{n})=0 \quad\text{or}\quad f(e_{n})=1\quad (n\in {\mathbb{N}}). \end{aligned}$$

If $f(e_{n})=0$ for all $n\in {\mathbb{N}}$ then $f\in h_{\varphi }^{\perp}$. This implies

$$\begin{aligned} f(x)^{2}=f\bigl(x^{2}\bigr)=0\quad (x\in \ell _{\varphi }) \end{aligned}$$

since $x^{2}\in h_{\varphi }$, see Proposition 3.1. Thus, $f=0$. If $f(e_{n})=f(e_{m})=1$ for some $n\neq m$, then

$$\begin{aligned} 0=f(e_{n}e_{m})=f(e_{n})f(e_{m})=1, \end{aligned}$$

a contradiction. Consequently, the only nonzero multiplicative functionals are the evaluation ones. □

Let A be a Banach algebra and let $\lambda \in \Phi _{A}\cup \{0\}$. A Banach A-bimodule X is called left λ-linked, resp. right λ-linked, if

$$\begin{aligned} a\cdot x=\langle a,\lambda \rangle x\quad (a\in A, x\in X), \end{aligned}$$

resp.

$$\begin{aligned} x\cdot a=\langle a,\lambda \rangle x \quad(a\in A, x\in X). \end{aligned}$$

A is said to be left λ-amenable, resp. right λ-amenable, if for any left λ-linked, resp. right λ-linked, Banach A-bimodule X every continuous derivation $\delta \colon A\to X'$ is inner. A is said to be λ-amenable if it is both left and right λ-amenable. A is said to be left character amenable, resp. right character amenable, if it is left λ-amenable, resp. right λ-amenable, for every $\lambda \in \Phi _{A}\cup \{0\}$. A more detailed account on character amenability can be found in [27, Sect. 4.3].

Remark 3.7

Let φ be an Orlicz function such that $a_{\varphi }=0$ and let $A\in \{\ell _{\varphi },h_{\varphi }\}$. From Proposition 3.6 and [10, Theorem 1.1] (cf. [27, Theorem 4.3.5]) it follows that A is λ-amenable for every $\lambda \in \Phi _{A}$. From [27, Theorem 4.3.4] it follows that A is never 0-amenable. Consequently, A is never character amenable.

4 Main results

Before proceeding with the main results we recall the following well-known facts from the theory of Orlicz functions. Let φ be an Orlicz function such that $a_{\varphi }=0$ and let $\ell _{\varphi }$ be the Orlicz sequence space. Obviously, φ is continuous and increasing on the interval $[0,b_{\varphi })$. Consequently, $\varphi ^{-1}$ is continuous and increasing on the interval $[0,\varphi (b_{\varphi }))$. Thus, $\varphi ^{-1}(\varphi (u))=u$ for all $u\in [0,b_{\varphi })$ and $\varphi (\varphi ^{-1}(v))=v$ for all $v\in [0,\varphi (b_{\varphi }))$. Then, a straightforward computation shows that

$$\begin{aligned} \exists k\in {\mathbb{N}} \ \forall n\geq k\colon \quad\Biggl\Vert \sum _{j=1}^{n}e_{j} \Biggr\Vert _{\varphi }= \frac{1}{\varphi ^{-1}(\frac{1}{n})}. \end{aligned}$$

(3)

In such a case, we will say that the equality (3) is satisfied for large $n\in {\mathbb{N}}$.

We will also be using the following notation. If $(a_{n})_{n\in {\mathbb{N}}}$ and $(b_{n})_{n\in {\mathbb{N}}}$ are two sequences of nonnegative numbers, then $(a_{n})_{n\in {\mathbb{N}}}\approx (b_{n})_{n\in {\mathbb{N}}}$ means that there is a constant $C>0$ such that

$$\begin{aligned} C^{-1}a_{n}\leq b_{n}\leq Ca_{n}\quad \text{for all }n\in {\mathbb{N}}. \end{aligned}$$

Recall that a Banach algebra A:

• (1) is called biprojective if the product map $\pi _{A}$ has a right inverse bimodule map;

• (2) is called biflat if the dual map $\pi _{A}'$ has a left inverse bimodule map;

• (3) has the π-property if $\pi _{A}(A\widehat{\otimes}A)=\overline{A^{2}}$.

Theorem 4.1

Let $\ell _{\varphi }$ be an Orlicz sequence algebra. TFAE:

• (i) $\ell _{\varphi }$ is biflat;

• (ii) $\ell _{\varphi }$ has the π-property;

• (iii) φ is not an N-function at 0.

Remark 4.2

Recall that if φ is not an N-function at 0 then either $a_{\varphi }>0$ (i.e., $\ell _{\varphi }=\ell _{\infty}$ with equivalent norms) or $\lim_{u\rightarrow 0}\frac{\varphi (u)}{u}>0$ (i.e., $\ell _{\varphi }=\ell _{1}$ with equivalent norms). Examples of Orlicz functions with the above properties will be provided in Example 4.8.

Proof

$(i)\Rightarrow (ii)$: Let $\pi \colon \ell _{\varphi }\widehat{\otimes}\ell _{\varphi }\to \ell _{\varphi }$ be the product map and let $\sigma \colon (\ell _{\varphi }\widehat{\otimes}\ell _{\varphi })' \to \ell _{\varphi }'$ be the left module inverse to $\pi '$. We will show that imπ is closed. To this end, let

$$\begin{aligned} \lim_{n\to \infty}\pi 'f_{n}=F \end{aligned}$$

for some sequence $(f_{n})_{n\in {\mathbb{N}}}\subset \ell _{\varphi }'$. Then,

$$\begin{aligned} \lim_{n\to \infty}f_{n}=\lim_{n\to \infty}\sigma \pi 'f_{n}=\sigma F. \end{aligned}$$

Therefore,

$$\begin{aligned} \pi '\sigma F=\lim_{n\to \infty}\pi 'f_{n}=F. \end{aligned}$$

Consequently, $F\in \operatorname {im}\pi '$ and the latter space is closed. From [19, Theorem 9.4] it follows that imπ is closed as well. However, then

$$\begin{aligned} \operatorname {im}\pi =\overline{\operatorname {im}\pi}=\overline{\ell _{\varphi }^{2}}. \end{aligned}$$

From [3, p. 166] it now follows that $\ell _{\varphi }$ has the π-property.

$(ii)\Rightarrow (iii)$: Assume that $\ell _{\varphi }$ has the π-property. If

$$\begin{aligned} a_{\psi}=\lim_{u\to 0^{+}}\frac{\varphi (u)}{u}>0, \end{aligned}$$

where ψ denotes the complementary function of φ (see formula (1)), then $\ell _{\varphi }=\ell _{1}$ (equivalent norms). Let us therefore assume that

$$\begin{aligned} a_{\psi}=\lim_{u\to 0^{+}}\frac{\varphi (u)}{u}=0. \end{aligned}$$

(4)

We will now show that $a_{\varphi }>0$ that is $\ell _{\varphi }=\ell _{\infty}$ as sets and the norms $\|\cdot \|_{\varphi }$ and $\|\cdot \|_{\infty}$ are equivalent. By assumption

$$\begin{aligned} \hat{\pi}\colon \ell _{\varphi }\widehat{\otimes}\ell _{\varphi }/ \ker \pi \to h_{\varphi } \end{aligned}$$

is an isomorphism. Since $h_{\varphi }'=\ell _{\psi}$ (see [16, Proposition 4.b.1]) and $(\ell _{\varphi }\widehat{\otimes}\ell _{\varphi }/\ker \pi )'=( \ker \pi )^{\perp}$ we obtain that

$$\begin{aligned} \hat{\pi}'\colon \ell _{\psi}\to (\ker \pi )^{\perp} \end{aligned}$$

(5)

is an isomorphism as well. Let

$$\begin{aligned} B_{j}\colon \ell _{\varphi }\times \ell _{\varphi }\to { \mathbb{K}},\qquad B_{j}(x,y):=x(j)y(j)\quad (j\in {\mathbb{N}}) \end{aligned}$$

be a continuous bilinear form. Clearly, $B_{j}\in \ker \pi ^{\perp}$. Moreover,

$$\begin{aligned} \bigl\langle u+\ker \pi,\hat{\pi}'e_{j}\bigr\rangle = \bigl\langle \pi (u),e_{j} \bigr\rangle =\langle u+\ker \pi,B_{j}\rangle \quad(u\in \ell _{ \varphi }\widehat{\otimes}\ell _{\varphi }). \end{aligned}$$

Therefore,

$$\begin{aligned} \hat{\pi}'e_{j}=B_{j}\quad \text{or}\quad \bigl(\hat{ \pi}'\bigr)^{-1}(B_{j})=e_{j}\quad (j\in { \mathbb{N}}). \end{aligned}$$

If we now denote

$$\begin{aligned} p_{n}:=\sum_{j=1}^{n}e_{j},\qquad \widehat{B_{n}}:=\sum_{j=1}^{n}B_{j}\quad (n\in {\mathbb{N}}), \end{aligned}$$

then

$$\begin{aligned} \bigl(\hat{\pi}'\bigr)^{-1}(\widehat{B_{n}})=p_{n}\quad (n\in {\mathbb{N}}). \end{aligned}$$

From (5) we obtain a constant $D>0$ such that

$$\begin{aligned} \Vert p_{n} \Vert _{\psi}\leq D \Vert \widehat{B_{n}} \Vert \quad (n\in {\mathbb{N}}), \end{aligned}$$

(6)

where on the right-hand side we consider the norm of a bilinear form, i.e., if $B\colon X\times Y\to Z$ is a bilinear mapping and $X,Y,Z$ are Banach spaces, then $\|B\|:=\sup \{\|B(x,y)\|_{Z}\colon \|x\|_{X}=\|y\|_{Y}=1\}$. Let us now compute these norms. From (3) and (4) it follows that

$$\begin{aligned} \Vert p_{n} \Vert _{\psi}=\frac{1}{\psi ^{-1}(\frac{1}{n})} \quad( \text{large }n\in {\mathbb{N}}). \end{aligned}$$

As for the other norms, let us first observe that

$$\begin{aligned} \Vert \widehat{B_{n}} \Vert = \Vert p_{n} \Vert _{{\mathcal{M}}(\ell _{\varphi },\ell _{ \psi})}\quad (n\in {\mathbb{N}}), \end{aligned}$$

where ${\mathcal{M}}(\ell _{\varphi },\ell _{\psi})$ is the multiplier space. From [4, Theorem 3] it now follows that

$$\begin{aligned} {\mathcal{M}}(\ell _{\varphi },\ell _{\psi})\simeq \ell _{\tau} \quad(\text{equivalent norms}), \end{aligned}$$

where τ is the Orlicz function defined by

$$\begin{aligned} \tau (s):=\max \bigl\{ 0,\sup \bigl\{ \psi (st)-\varphi (t)\colon t\in [0,1] \bigr\} \bigr\} . \end{aligned}$$

Therefore, we obtain that $(\|\widehat{B_{n}}\| )_{n\in {\mathbb{N}}}\approx (\|p_{n} \|_{\tau} )_{n\in {\mathbb{N}}}$. Assume for the moment that $a_{\tau}>0$, i.e., $\ell _{\tau}=\ell _{\infty}$ (equivalent norms). Then, there exist constants $D_{1},D_{2}>0$ such that

$$\begin{aligned} D_{1}\leq \Vert p_{n} \Vert _{\tau}\leq D_{2}\quad (n\in {\mathbb{N}}). \end{aligned}$$

In particular, condition (6) takes the form

$$\begin{aligned} \frac{1}{\psi ^{-1}(\frac{1}{n})}\leqslant D_{3} \end{aligned}$$

for some constant $D_{3}>0$ and all large $n\in {\mathbb{N}}$. Equivalently (see the discussion at the beginning of this section),

$$\begin{aligned} \psi \biggl(\frac{1}{D_{3}} \biggr)\leqslant \frac{1}{n} \xrightarrow[n\to \infty ]{}0, \end{aligned}$$

which contradicts the fact that $a_{\psi}=0$. Therefore, $a_{\tau}=0$ and

$$\begin{aligned} \Vert p_{n} \Vert _{\tau}=\frac{1}{\tau ^{-1}(\frac{1}{n})} \quad( \text{large }n\in {\mathbb{N}}). \end{aligned}$$

Coming back to (6) we obtain another constant $C>0$ such that

$$\begin{aligned} \tau ^{-1} \biggl(\frac{1}{n} \biggr)\leq C\psi ^{-1} \biggl(\frac{1}{n} \biggr) \quad(\text{large }n\in {\mathbb{N}}). \end{aligned}$$

(7)

The convexity of ψ now implies that

$$\begin{aligned} \psi \bigl(t\psi ^{-1}(s)\bigr)-\varphi (t)\leq ts-\varphi (t)\quad \bigl(t \in [0,1],s \geq 0\bigr). \end{aligned}$$

Recall that from [16, Proposition 4.a.5] it follows that we may assume that $b_{\psi}=\infty$ that then implies that $\psi ^{-1}$ is well defined for every $s\geq 0$. Hence,

$$\begin{aligned} \sup \bigl\{ \psi \bigl(t\psi ^{-1}(s)\bigr)-\varphi (t)\colon t\in [0,1]\bigr\} \leq \sup \bigl\{ ts-\varphi (t)\colon t\geq 0\bigr\} =\psi (s). \end{aligned}$$

Consequently,

$$\begin{aligned} \tau \circ \psi ^{-1}\leq \psi \end{aligned}$$

or, equivalently,

$$\begin{aligned} \psi ^{-1}\leq \tau ^{-1}\circ \psi. \end{aligned}$$

(8)

If we denote $u_{n}:=\psi ^{-1}(\frac{1}{n})$ (equivalently, $\psi (u_{n})=\frac{1}{n}$, see the discussion at the beginning of this section) then (7) and (8) imply that

$$\begin{aligned} \psi ^{-1}(u_{n})\leqslant \tau ^{-1}\circ \psi (u_{n})\leqslant Cu_{n}\quad (\text{large }n\in {\mathbb{N}}), \end{aligned}$$

whence

$$\begin{aligned} u_{n}\leq \psi (Cu_{n}) \quad(\text{large }n\in {\mathbb{N}}). \end{aligned}$$

Equivalently,

$$\begin{aligned} \frac{\psi (Cu_{n})}{Cu_{n}}\geq \frac{1}{C}\quad (\text{large }n \in {\mathbb{N}}). \end{aligned}$$

We now recall that $u_{n}=\psi ^{-1}(\frac{1}{n})\xrightarrow[n\to \infty ]{}0$, which implies that

$$\begin{aligned} a_{\varphi }=\lim_{u\to 0^{+}}\frac{\psi (u)}{u}\geq \frac{1}{C}>0. \end{aligned}$$

Consequently, $\ell _{\varphi }=\ell _{\infty}$ as sets and the respective norms are equivalent.

$(iii)\Rightarrow (i)$: By assumption, the Orlicz algebra $\ell _{\varphi }$ is either $\ell _{\infty}$ (in the case where $a_{\varphi }>0$) or $\ell _{1}$ (in the case where $\lim_{u\to 0^{+}}\varphi (u)/u>0$). That $\ell _{\infty}$ is biflat follows from [28] (cf. [3, Theorem 2.9.65]) since $\ell _{\infty}$ is a commutative $C^{*}$-algebra, therefore amenable by [9, Lemma 7.10]. As for the other case observe that $\ell _{1}$ is even biprojective—see [3, Example 4.1.42]. □

Corollary 4.3

Let $\ell _{\varphi }$ be an Orlicz sequence algebra. TFAE:

• (i) $\ell _{\varphi }$ is biprojective;

• (ii) $\ell _{\varphi }=\ell _{1}$ (equivalent norms).

Proof

That $\ell _{1}$ is biprojective follows from [3, Example 4.1.42]. Assume now that $\ell _{\varphi }$ is biprojective. It is therefore biflat and Theorem 4.1 implies that it is either $\ell _{\infty}$ or $\ell _{1}$. That $\ell _{\infty}$ is not biprojective follows from [3, Theorem 2.8.48] and [27, Corollary 4.1.5]. □

Recall that an approximate identity in a Banach algebra A is a net $(e_{\alpha})_{\alpha \in \Lambda}\subset A$ such that

$$\begin{aligned} \lim_{\alpha}e_{\alpha}a=a=\lim_{\alpha}ae_{\alpha}\quad (a\in A). \end{aligned}$$

It is called sequential if Λ is countable and bounded if the set $\{e_{\alpha}\colon \alpha \in \Lambda \}$ is bounded in A.

A Banach algebra A is called:

• (1) essential if $\overline{A^{2}}=A$ as sets;

• (2) weakly amenable if every continuous derivation $\delta \colon A\to A'$ is inner;

• (3) approximately semicontractible if for any A-bimodule X and every continuous derivation $\delta \colon A\to X$ there are nets $(x_{\alpha})_{\alpha \in \Lambda}, (y_{\alpha})_{\alpha \in \Lambda} \subset X$ such that

  $$\begin{aligned} \delta (a)=\lim_{\alpha}(a\cdot x_{\alpha}-y_{\alpha} \cdot a)\quad (a \in A). \end{aligned}$$

  If, in addition we can always choose $x_{\alpha}=y_{\alpha}, \alpha \in \Lambda$ then A is approximately contractible. If, moreover, the net $(\operatorname {ad}_{x_{\alpha}})_{\alpha \in \Lambda}$ is bounded in ${\mathcal{B}}(A,X)$ then A is boundedly approximately contractible. If the above properties hold only for dual A-bimodules then A is said to be (boundedly) approximately (semi-)amenable.

A more detailed account on (bounded) approximate (semi-)amenability/contractibility can be found in [27, Sect. 4.4].

Now, we are ready to prove the following characterization.

Theorem 4.4

Let $\ell _{\varphi }$ be an Orlicz sequence algebra. TFAE:

• (i) $\ell _{\varphi }$ admits a (sequential) approximate identity;

• (ii) $\ell _{\varphi }$ is essential;

• (iii) $\ell _{\varphi }$ is weakly amenable;

• (iv) either $\varphi \in \Delta _{2}(0)$ or $a_{\varphi }>0$;

• (v) $\ell _{\varphi }$ is approximately semiamenable.

Remark 4.5

Observe that in condition (iv) we have an “exclusive-or” relation, i.e., properties $\varphi \in \Delta _{2}(0)$ and $a_{\varphi }>0$ cannot hold simultaneously for any Orlicz function φ.

Proof

We will show two chains of implications, namely

$$\begin{aligned} (i)\quad\Rightarrow\quad (ii)\quad\Rightarrow\quad (iii)\quad\Rightarrow\quad (iv)\quad\Rightarrow\quad (i) \end{aligned}$$

and

$$\begin{aligned} (iv)\quad\Leftrightarrow\quad (v). \end{aligned}$$

$(i)\Rightarrow (ii)$: this implication is clear.

$(ii)\Rightarrow (iii)$: if $a_{\varphi }>0$ then $\ell _{\varphi }=\ell _{\infty}$ is even amenable by [9, Lemma 7.10] therefore weakly amenable. If $a_{\varphi }=0$ then from the assumption and Proposition 3.1 it follows that $\ell _{\varphi }=\overline{(\ell _{\varphi })^{2}}=h_{\varphi }$, therefore $(e_{n})_{n\in {\mathbb{N}}}$ is a Schauder basis in $\ell _{\varphi }$ consisting of idempotents. From [3, Proposition 2.8.72] the conclusion follows.

$(iii)\Rightarrow (iv)$: Suppose that $\varphi \notin \Delta _{2}(0)$ and $a_{\varphi }=0$. Since $\varphi \notin \Delta _{2}(0)$, we obtain that $h_{\varphi }\varsubsetneq \ell _{\varphi }$ (see Proposition [16, Proposition 4.a.4]). From the Hahn–Banach Theorem it now follows that there exists a continuous nonzero functional $g\in \ell _{\varphi }'$ that vanishes on $h_{\varphi }$. Then, the mapping

$$\begin{aligned} \delta \colon \ell _{\varphi }\to \ell _{\varphi }',\quad \delta (a):=g(a)g \end{aligned}$$

defines a nonzero continuous derivation. Indeed,

$$\begin{aligned} \bigl\Vert \delta (a) \bigr\Vert = \bigl\vert g(a) \bigr\vert \Vert g \Vert \leq \Vert g \Vert ^{2} \Vert a \Vert _{\varphi } \quad(a \in \ell _{\varphi }), \end{aligned}$$

where for any $f\in \ell _{\varphi }'$ we have $\|f\|=\sup \{|f(a)|\colon \|a\|_{\varphi }=1\}$. Moreover, since $a_{\varphi }=0$, from Proposition 3.1 it follows that

$$\begin{aligned} \delta (ab)=g(ab)g=0 \quad(a,b\in \ell _{\varphi }) \end{aligned}$$

and

$$\begin{aligned} \bigl\langle c,a\cdot \delta (b)\bigr\rangle =g(b)g(ca)=0,\qquad \bigl\langle c, \delta (a)\cdot b\bigr\rangle =g(a)g(bc)=0\quad (a,b,c\in \ell _{\varphi }). \end{aligned}$$

Clearly, δ is nonzero since for $a\notin \ker g$ we have

$$\begin{aligned} \bigl\langle a,\delta (a)\bigr\rangle =g(a)^{2}\neq 0. \end{aligned}$$

Since $\ell _{\varphi }$ is commutative the only inner derivation is the trivial one. Thus, δ is not inner that contradicts the weak amenability of $\ell _{\varphi }$. Consequently, $\varphi \in \Delta _{2}(0)$ or $a_{\varphi }>0$.

$(iv)\Rightarrow (i)$: if $a_{\varphi }>0$ then $\ell _{\varphi }=\ell _{\infty}$ (equivalent norms) is even unital and if $\varphi \in \Delta _{2}(0)$ then it follows that $\ell _{\varphi }=h_{\varphi }$ (see [16, Proposition 4.a.4]) and $p_{n}:=\sum_{j=1}^{n}e_{j}$ is a sequential approximate identity.

$(iv)\Rightarrow (v)$: if $a_{\varphi }>0$ then $\ell _{\varphi }=\ell _{\infty}$ (equivalent norms) is even amenable by [9, Lemma 7.10] therefore approximately semiamenable. Let $\varphi \in \Delta _{2}(0)$. For any $n\in {\mathbb{N}}$ let $E_{n}:=(p_{n}\cdot \ell _{\varphi },\|\cdot \|_{\varphi })$ and let X be an $\ell _{\varphi }$-bimodule. If

$$\begin{aligned} \delta \colon \ell _{\varphi }\to X \end{aligned}$$

is a continuous derivation, then X is (in a natural way) an $E_{n}$-bimodule and

$$\begin{aligned} \delta _{n}\colon E_{n}\to X,\qquad \delta _{n}(a):= \delta (a) \end{aligned}$$

is also a continuous derivation. Since $E_{n}$ is finite dimensional and semisimple from [3, Theorem 1.9.21] it follows that $\delta _{n}$ is inner, i.e., there is $\xi _{n}\in X$ such that

$$\begin{aligned} \delta _{n}(a)=a\cdot \xi _{n}-\xi _{n}\cdot a\quad (a\in E_{n}, n \in {\mathbb{N}}). \end{aligned}$$

By assumption $\ell _{\varphi }=h_{\varphi }$, therefore

$$\begin{aligned} a=\lim_{n}p_{n}a \quad\text{and}\quad p_{n}a\in E_{n} \end{aligned}$$

and

$$\begin{aligned} \delta (a)=\lim_{n}\delta _{n}(p_{n}a)= \lim_{n}(p_{n}a\cdot \xi _{n}- \xi _{n}\cdot p_{n}a)=\lim_{n}\bigl(a\cdot (p_{n}\cdot \xi _{n})-(p_{n} \cdot \xi _{n})\cdot a\bigr). \end{aligned}$$

Consequently, δ is approximately semiinner.

$(v)\Rightarrow (iv)$: suppose that $\varphi \notin \Delta _{2}(0)$ and $a_{\varphi }=0$ and define

$$\begin{aligned} \delta \colon \ell _{\varphi }\to \ell _{\varphi }/h_{\varphi },\qquad \delta (a):=a+h_{\varphi }, \end{aligned}$$

where $\ell _{\varphi }/h_{\varphi }$ is the quotient module. From Proposition 3.1 it follows that the module actions in this quotient module are trivial and that δ is a derivation. If it were to be approximately semiinner then it would have to be trivial. However, $\delta (a)\neq 0$ for any $a\notin h_{\varphi }$. □

Recall that a Banach algebra A is said to be pseudoamenable if there is a (possibly unbounded) net $(d_{\alpha})_{\alpha}\subset A\widehat {\otimes }A$ such that

$$\begin{aligned} a\cdot d_{\alpha}-d_{\alpha}\cdot a\xrightarrow[\alpha ]{}0 \quad\text{and}\quad a\pi (d_{\alpha})\xrightarrow[\alpha ]{}0\quad (a\in A). \end{aligned}$$

A more detailed account on pseudoamenability can be found in [27, Sect. 4.4].

Remark 4.6

(1) From [1, Lemma 2.4 and Theorem 2.5] it follows that $\ell _{\varphi }$ as well as $h_{\varphi }$ are never approximately amenable unless $a_{\varphi }>0$. Indeed, $(p_{n})_{n\in {\mathbb{N}}}\subset h_{\varphi }$ constitutes an unbounded but multiplier bounded sequence satisfying $p_{n}p_{n+1}=p_{n}=p_{n+1}p_{n}, n\in {\mathbb{N}}$.

(2) From [5, Theorem 13] it follows that if $\varphi \in \Delta _{2}(0)$ then $\ell _{\varphi }$ is even boundedly approximately contractible.

(3) From [6, Corollary 3.7] and [27, Proposition 4.4.2] it follows that $h_{\varphi }$ is always pseudoamenable, whereas $\ell _{\varphi }$ is pseudoamenable if and only if $\varphi \in \Delta _{2}(0)$ or $a_{\varphi }>0$.

Remark 4.7

Condition (iv) of the above theorem shows in particular that there is a wide class of Orlicz sequence algebras (i.e., $a_{\varphi }=0$ and $\varphi \notin \Delta _{2}(0)$) that serve as nonexamples to a number of amenability properties.

We end this section with a list of examples illustrating properties of Orlicz functions considered in Theorems 4.1 and 4.4.

Example 4.8

(1) The Orlicz functions

$$\begin{aligned} \varphi _{1}(u):=\max \bigl\{ 0,u^{p}-1\bigr\} \quad (p\geqslant 1) \end{aligned}$$

and

$$\begin{aligned} \varphi _{2}(u):= \textstyle\begin{cases} 0, & u\in [0,1], \\ \infty, & u\in (1,\infty ), \end{cases}\displaystyle \end{aligned}$$

satisfy $a_{\varphi _{1}}=a_{\varphi _{2}}=1$.

(2) The Orlicz functions

$$\begin{aligned} \varphi _{a,b,c}(u):=au\ln (b+cu)\quad (a>0,b>1,c\geqslant 0) \end{aligned}$$

are not N-functions at 0. Clearly, if φ is an Orlicz function such that $a_{\varphi }=0$ and φ is not an N-function at 0 then it satisfies condition $\Delta _{2}(0)$.

(3) The Orlicz functions

$$\begin{aligned} \varphi _{3}(u):=u^{p}\quad (p>1) \end{aligned}$$

and

$$\begin{aligned} \varphi _{4}(u):=u^{p}\ln (1+u) \quad(p\geqslant 1) \end{aligned}$$

are N-functions at 0 and satisfy condition $\Delta _{2}(0)$.

(4) We finish with a construction of an Orlicz function for which $a_{\varphi }=0$ but it does not satisfy condition $\Delta _{2}(0)$. Let

$$\begin{aligned} u_{n}:=\frac{1}{2^{n}}\quad \bigl(n\in {\mathbb{N}}\cup \{0\}\bigr) \end{aligned}$$

and let $p\colon [0,\infty )\to [0,\infty )$ be a function defined as

$$\begin{aligned} p(0):=0,\qquad p(t):=\frac{1}{n!}\quad\text{for $t\in [u_{n},u_{n-1})$}, n\in {\mathbb{N}},\qquad p(t):=t\quad \text{for $t\geqslant 1$}. \end{aligned}$$

The function p is nondecreasing and right-continuous. Now, we define the Orlicz function $\varphi _{5}$ as

$$\begin{aligned} \varphi _{5}(u):= \int _{0}^{u}p(t)\,dt. \end{aligned}$$

Clearly, $a_{\varphi _{5}}=0$. Moreover, for any $n\in {\mathbb{N}}$ we have

$$\begin{aligned} \varphi _{5}(2u_{n}) & = \int _{0}^{2u_{n}}p(t)\,dt \\ & \geqslant \int _{u_{n}}^{2u_{n}}\frac{1}{n!}\,dt \\ & =(n+1) \int _{0}^{u_{n}}\frac{1}{(n+1)!}\,dt \\ & \geqslant (n+1) \int _{0}^{u_{n}}p(t)\,dt=(n+1)\varphi _{5}(u_{n}). \end{aligned}$$

Therefore, the function $\varphi _{5}$ does not satisfy condition $\Delta _{2}(0)$.