Arens regularity of certain weighted semigroup algebras and countability

Abstract

In this paper, among other things, we show that for a large class of semigroups, the Arens regularity of the weighted semigroup algebra $\ell _1(S,\omega )$ implies the countability of S. This generalizes the main result of Craw and Young (Q J Math 25:351–358, 1974), stating that every countable semigroup admits a weight $\omega$ for which $\ell _1(S,\omega )$ is Arens regular and no uncountable group admits such a weight.

1 Introduction

Arens [2] introduced two (Arens) multiplications on the second dual of a Banach algebra turning it into a Banach algebra. A Banach algebra is said to be Arens regular if the Arens multiplications coincide on its second dual. The Arens regularity of the semigroup algebra $\ell _1(S)$ and the weighted semigroup algebra $\ell _1(S,\omega )$ have been investigated in [7] and [3, 4], respectively. Recent developments on the Arens regularity of $\ell _1(S,\omega )$ are presented in [5].

In this paper we first show that the Arens regularity of a weighted semigroup algebra is stable under certain homomorphisms of semigroups (Proposition 3.1). Then we study those semigroups for which the Arens regularity of $\ell _1(S,\omega )$ necessities the countability of S (a known example for such a semigroup is actually a group; see [4]). As the main aim of the paper we shall show that for a wide variety of semigroups the Arens regularity of $\ell _1(S,\omega )$ implies that S is countable; (Theorem 4.5 infra).

2 Preliminaries

Let S be a semigroup and $\omega : S\rightarrow (0,\infty )$ be a weight on S, i.e. $\omega (st)\le \omega (s)\omega (t)$ for all $s,t\in S$. Then we define the mapping $\Omega : S\times S\rightarrow (0,1]$ by $\Omega (s,t)=\frac{\omega (st)}{\omega (s)\omega (t)},\quad (s,t\in S)$.

Definition 2.1

Let X, Y be sets and f be a complex-valued function on $X\times Y$.

• (1) We say that f is cluster on $X\times Y$ if for each pair of sequences $(x_n)$, $(y_m)$ of distinct elements of X, Y, respectively

  $$\begin{aligned} \lim _n\lim _mf(x_n,y_m)=\lim _m\lim _nf(x_n,y_m) \end{aligned}$$

  (2.1)

  whenever both sides of (2.1) exist.

• (2) If f is cluster and both sides of 2.1 are zero (respectively positive) in all cases, we say that f is 0-cluster (respectively positive cluster).

For a weight $\omega$ on a semigroup S, 

$$\begin{aligned} \ell _1(S,\omega ):=\left\{ f:S\rightarrow \mathbb {C}: \Vert f\Vert _{\omega ,1}=\sum _{s\in S}|f(s)|\omega (s)<\infty \right\} \end{aligned}$$

is a Banach algebra under the pointwise linear space operations, the convolution multiplication and the norm $\Vert \cdot \Vert _{\omega ,1}$. Many aspects of the weighted semigroup algebra $\ell _1(S,\omega )$ are investigated in the extensive memoir [5]. For ease of reference we quote the following criterion from [3] which will be frequently used in the sequel.

Theorem 2.2

([3, Lemma 3.1 and Theorem 3.3]) For a weighted semigroup algebra $\ell _1(S,\omega ),$ the following statements are equivalent.

• (i) $\ell _1(S,\omega )$ is regular.

• (ii) The map $(x,y)\mapsto h(xy)\Omega (x,y)$ is cluster on $S\times S$ for every bounded function h on S.

• (iii) For each pair of sequences $(x_n),(y_m)$ of distinct points of S there exist subsequences $(a_n),(b_m)$ of $(x_n),(y_m)$, respectively, such that

  • either $\lim _n\lim _m\Omega (a_n, b_m)=0=\lim _m\lim _n\Omega (a_n, b_m),$

  • or the matrix $(a_nb_m)$ is of type C (i.e. either the rows or the columns of $(a_nb_m)$ are constant and distinct).

In particular, if $\Omega$ is 0-cluster then $\ell _1(S,\omega )$ is regular.

3 Arens regularity of $\ell _1(S,\omega )$ and homomorphisms

Let $\psi : S\rightarrow T$ be a homomorphism of semigroups. If $\omega$ is a weight on $T$, then trivially $\overleftarrow{{\omega }}(s):=\omega (\psi (s))$ defines a weight on S.

If $\psi : S\rightarrow T$ is an epimorphism and $\omega$ is a bounded below (that is, $\inf \omega (S)>0$) weight on S then a direct verification reveals that $\overrightarrow{{\omega }}(t):=\inf \omega (\psi ^{-1}(t))$ defines a weight on T.

We present the next elementary result concerning to the stability of regularity under homomorphisms.

Proposition 3.1

Let $\psi : S\rightarrow T$ be a homomorphism of semigroups.

• (i) If $\psi$ is onto and $\omega$ is a bounded below weight on S then the regularity of $\ell _1(S,\omega )$ necessities the regularity of $\ell _1(T,\overrightarrow{{\omega }})$. Furthermore, if $\Omega$ is 0-cluster, then so is $\overrightarrow{{\Omega }}$.

• (ii) For a weight $\omega$ on T if $\ell _1(S,\overleftarrow{{\omega }})$ is regular, then $\ell _1(T,\omega )$ is regular.

Proof

To prove (i), since $\omega$ is bounded below, we can assume that, $\inf \omega (S)\ge \varepsilon >0$, for some $\varepsilon <1$, so $\overrightarrow{{\omega }}\ge \varepsilon$. Let $(x_n),(y_m)$ be sequences of distinct elements in T. Then there are sequences of distinct elements $(s_n), (t_m)$ in S such that

$$\begin{aligned} \left\{ \begin{array}{lr} \overrightarrow{{\omega }}(x_n) >\omega (s_n)(1-\varepsilon ) &{} \text{ and }\quad \psi (s_n)=x_n, \\ \overrightarrow{{\omega }}(y_m) >\omega (t_m)(1-\varepsilon ) &{} \text{ and }\quad \psi (t_m)=y_m. \end{array}\right. \end{aligned}$$

It follows that $\overrightarrow{{\omega }}(x_n) \overrightarrow{{\omega }}(y_m)>\omega (s_n)\omega (t_m)(1-\varepsilon )^2$ and so from $\overrightarrow{{\omega }}(x_ny_m)\le \omega (s_nt_m)$ we get $\frac{\overrightarrow{{\omega }}(x_ny_m)}{\overrightarrow{{\omega }}(x_n) \overrightarrow{{\omega }}(y_m)}\le \frac{1}{(1-\varepsilon )^2}\frac{ \omega (s_nt_m)}{\omega (s_n)\omega (t_m)};$ or equivalently,

$$\begin{aligned} \overrightarrow{{\Omega }}(x_n,y_m)\le \frac{1}{(1-\varepsilon )^2}\Omega (s_n,t_m),\quad (n,m\in {\mathbb {N}}). \end{aligned}$$

(3.1)

Applying the inequality (3.1), a standard argument based on part (iii) of Theorem 2.2 shows that if $\ell _1(S,\omega )$ is regular then $\ell _1(T,\overrightarrow{{\omega }})$ is regular.

To prove (ii), let h be a bounded function on T. As $\ell _1(S,\overleftarrow{{\omega }})$ is regular, by Theorem 2.2, the mapping $(s,t)\rightarrow h\circ \psi (st)\overleftarrow{{\Omega }}(s,t)$ is cluster on $S\times S$. It follows that the mapping $(x,y)\mapsto h(xy)\Omega (x,y)$ is cluster on $T\times T$ which implies that $\ell _1(T,\omega )$ is regular.$\square$

Corollary 3.2

Let $\psi :S\rightarrow T$ be a homomorphism of semigroups. If $\ell _1(S)$ is Arens regular then $\ell _1(T,\omega )$ is Arens regular, for every weight function $\omega$ on T.

Proof

Let $\ell ^1(S)$ be Arens regular and let $\omega$ be a weight on T. Then $\ell ^1(S,\overleftarrow{\omega })$ is Arens regular by [3, Corollary3.4]. Proposition 3.1 now implies that $\ell _1(T,\omega )$ is Arens regular. $\square$

4 Arens regularity of $\ell _1(S,\omega )$ and the countability of S

We commence with the following elementary lemma that will be used frequently in the sequel.

Lemma 4.1

A nonempty set X is countable if and only if there exists a function $f:X\rightarrow (0,\infty )$ such that the sequence $(f(x_n))$ is unbounded for every sequence $(x_n)$ of distinct elements in X.

Proof

If $X=\{x_n:n\in {\mathbb {N}}\}$ is countable the $f(x_n)=n$ is the desired function. For the converse, suppose that such a function $f:X\rightarrow (0,\infty )$ exists. Since $X=\cup _{n\in \mathbb N}\{x\in X:f(x)\le n \}$ and each of the sets $\{x\in X:f(x)\le n\}$ is countable, so X is countable. $\square$

Applying the latter lemma we give the next result of [4] with a slightly simpler proof.

Corollary 4.2

(See [4, Corollary1]) On every countable semigroup S there exists a bounded below weight $\omega$ such that $\Omega$ is 0-cluster. In particular, $\ell _1(S,\omega )$ is Arens regular.

Proof

Let F be the free semigroup generated by the countable semigroup $S=\{a_k:\quad k\in {\mathbb {N}}\}$. For every element $x\in F$ (with the unique presentation $x=a_{k_1}a_{k_2}\cdots a_{k_r}$) set $\omega _1(x)=1+k_1+k_2+\cdots k_r$. A direct verification shows that $\omega _1$ is a weight on F with $1\le \omega _1,$ and that $\Omega _1$ is 0-cluster. Let $\psi : F\rightarrow S$ be the canonical epimorphism. Set $\omega :=\overrightarrow{{\omega _1}}.$ By Proposition 3.1, $\omega$ is our desired weight on S. $\square$

The next result is a converse to the Corollary 4.2.

Proposition 4.3

If there exists a bounded below weight $\omega$ on S such that $\Omega$ is 0-cluster, then S is countable.

Proof

Let $\omega$ be a bounded below weight for which $\Omega$ is 0-cluster. Let $\epsilon >0$ satisfy $\omega \ge \epsilon .$ Let S be uncountable. By Lemma 4.1 there is a sequence $(s_n)$ of distinct elements in S and $n_0\in {\mathbb {N}}$ such that $\omega (s_n)\le n_0$ for all $n\in {\mathbb {N}}$. For all subsequences $(s_{n_k}),(s_{m_l})$ of $(s_n)$ we get

$$\begin{aligned} \Omega (s_{n_k},s_{m_l})=\frac{\omega (s_{n_k}s_{m_l})}{\omega (s_{n_k})\omega (s_{m_l})}\ge \frac{\epsilon }{n_0^2},\quad (k,l\in {\mathbb {N}}), \end{aligned}$$

contradicting the 0-clusterity of $\Omega$. $\square$

Combining Corollary 4.2 and Proposition 4.3 we arrive to the next result.

Corollary 4.4

For every semigroup S the following assertions are equivalent.

• (1) There is a bounded below weight $\omega$ on S such that $\Omega$ is 0-cluster.

• (2) S is countable.

As it has been noted in Theorem 2.2, if $\Omega$ is 0-cluster then $\ell _1(S,\omega )$ is Arens regular; and, the converse is not true, in general. For example, $\ell _1(S)$ is Arens regular for every zero semigroup S,  but the weight $\omega =1$ is not 0-cluster. However, the converse holds in the case where S is weakly cancellative; (see [3, Corollary3.8]).

In the sequel, we replace the condition “$\Omega$ is 0-cluster” by “$\ell _1(S,\omega )$ is Arens regular” and investigate how it influences the countability of S. Indeed, as we shall see in the next result, for a wide variety of semigroups (including Brandt semigroups, (0-)simple inverse semigroups and inverse semigroups with finite set of idempotents; see [6]) the Arens regularity of the weighted semigroup algebra $\ell ^1(S, \omega )$ necessities the countability of S.

Theorem 4.5

For every Brandt semigroup (resp. completely [0-]simple semigroup, [0-]simple inverse semigroup, inverse semigroup with finite set of idempotents) S, the following statements are equivalent.

• (1) There is a bounded below weight $\omega$ on S such that $\ell _1(S,\omega )$ is Arens regular.

• (2) S is countable.

Proof

If S is countable then Corollary 4.2 guarantees the existence of a weight $\omega$ satisfying (1). For the converse, let $\omega$ be a weight on S such that $\ell _1(S,\omega )$ is Arens regular. Then in either of the following cases ((i), (ii), (iii)) we shall show that S is countable.

(i) Let $S=M^0(G,I,I,\Delta )$ be an infinite Brandt semigroup (see [6]), and let $\omega \ge \epsilon >0$. We show that $G\times I$ is countable. First, let $(i_n)$ be an arbitrary sequence of distinct elements in I and let $x_n=(i_n,1,i_1), y_m=(i_1,1, i_m)$. Then $x_n.y_m=(i_n,1,i_m)$ and so for any subsequences $(a_n),(b_m)$ of $(x_n),(y_m)$, respectively, the matrix $(a_nb_m)$ is not of type C, so by Theorem 2.2, $\lim _n\lim _m\Omega (a_n,b_m)=0=\lim _m\lim _n\Omega (a_n,b_m).$ But $\Omega (a_n,b_m)\ge \frac{\epsilon }{\omega (a_n)\omega (b_m)}$, for each m, n, and this implies that either $(\omega (a_n))$ or $(\omega (b_m))$ is unbounded. Let $(\omega (a_n))$ be unbounded, and define $f(i):=\omega (i,1,i_1)\quad (i\in I)$. By Lemma 4.1, I is countable.

Let $(s_n)$ be an arbitrary sequence of distinct elements in G, and let $x_n=(i_0,s_n,i_0), y_m=(i_0,s_m, i_0)$. Then $x_n.y_m=(i_0,s_ns_m,i_0)$ and for any subsequences $(a_n),(b_m)$ of $(x_n),(y_m)$, respectively, it is easy to verify that the matrix $(a_nb_m)$ is not of type C. Thus by Theorem 2.2, $\lim _n\lim _m\Omega (a_n,b_m)=0=\lim _m\lim _n\Omega (a_n,b_m).$ But $\Omega (a_n,b_m)\ge \frac{\epsilon }{\omega (a_n)\omega (b_m)}$, and this implies that either $(\omega (a_n))$ or $(\omega (b_m))$ is unbounded. Define $f(s):=\omega (i_0,s,i_0),\quad (s\in G)$. By Lemma 4.1, G is countable. By what we have shown the Brandt semigroup $S=M^0(G,I,I,\Delta )$ is countable.

(ii) Suppose that S is completely $0-$simple, then as it has been explained in [6], S has the presentation $S\cong M^0(G,I,\Lambda ;P)=(I\times G\times \Lambda )\cup \{0\}$, equipped with the multiplication

$$\begin{aligned} (i,a,\lambda )(j,b,\mu ) =\left\{ \begin{array}{cr} (i,ap_{\lambda j}b,\mu ) &{}\text{ if }\quad p_{\lambda j}\not =0\\ 0 &{}\,\, \text{ if }\quad p_{\lambda j}=0, \end{array}\right. \\ (i,a,\lambda )0=0(i,a,\lambda )=0. \end{aligned}$$

Fix $i_0\in I$, $\lambda _0\in \Lambda$ such that $p_{\lambda _0 i_0}\not =0$ and define $f:I\times \Lambda \rightarrow (0,\infty )$ by $f(i,\lambda )=\omega (i,1,\lambda _0)\omega (i_0,1,\lambda )$. Let $(i_n,\lambda _n)$ be a sequence of distinct elements in $I\times \Lambda$ and set $x_n= (i_n,1,\lambda _0) ,\quad y_m=(i_0,1,\lambda _m).$ It is readily verified that if $p_{\lambda _0i_n}\not =0$ then $x_nx_m=(i_n,p_{\lambda _0i_0},\lambda _m)$ for all $n,m\in {\mathbb {N}}$. So for any subsequences $(a_n),(b_m)$ of $(x_n),(y_m)$, respectively, the matrix $(a_nb_m)$ is not of type C. Thus by Theorem 2.2 $\lim _n\lim _m\Omega (a_n,b_m)=0=\lim _m\lim _n\Omega (a_n,b_m).$ But $\Omega (a_n,b_m)\ge \frac{\epsilon }{\omega (a_n)\omega (b_m)}$, and this implies $\omega (a_n)$ or $\omega (b_m)$ is unbounded. Thus $f(i_n,\lambda _n)=\omega (a_n)\omega (b_n)$ is unbounded sequence. Lemma 4.1 implies that $I\times \Lambda$ is countable.

We are going to show that G is also countable. Set $\omega _0(g)=\omega (i_0,gp_{\lambda _0,i_0}^{-1},\lambda _0)\quad (g\in G)$. Then $\omega _0$ is a weight on G such that $\Omega _0$ is 0-cluster and by Corollary 4.4 G is countable. Therefore S is countable as claimed. The case where S is completely simple needs a similar argument.

(iii) Let S be an inverse semigroup such that the set of idempotents in S is finite. By [6] there is a principal series

$$\begin{aligned} S=S_0\supsetneq S_1\supsetneq \cdots \supsetneq S_m=G\supsetneq S_{m+1}=\emptyset , \end{aligned}$$

for some group G, such that $S_i/S_{i+1}$ is a Brandt semigroup, for each $i,\quad (0\le i\le m)$. For each i set $\omega _i:=\overrightarrow{{\omega }_{|_{S_i}}}$ for the natural epimorphism $S_i\rightarrow S_i/S_{i+1}.$ By Proposition 3.1 $\ell _1(S_i/S_{i+1}, \omega _i)$ is Arens regular and now part (i) of the proof implies that $S_i/S_{i+1}$ is countable. We thus have that $S_i\setminus S_{i+1}$is countable, for $0\le i\le m$. Therefore $S=\bigcup _{i=0}^m(S_i\setminus S_{i+1})$ is countable, as required. Proof for the 0-simple inverse semigroup case is similar. $\square$

The next example illustrates that, a countable (Brandt) semigroup S may admits some weight $\omega$ for which $\ell _1(S,\omega )$ is not Arens regular.

Example 4.6

Let $S=M^0(\{e\},{\mathbb {N}},{\mathbb {N}},\Delta )$. Define $\omega :S\rightarrow [1,3]$ by

$$\begin{aligned} \omega (n,e,m)=1+\frac{1}{n}+\frac{1}{m},\quad \omega (0)=1. \end{aligned}$$

Let $x_n=(n,e,1), y_m=(1,e, m)$. Then $x_n.y_m=(n,e,m)$ and so for any subsequences $(a_n),(b_m)$ of $(x_n),(y_m)$, respectively, the matrix $(a_nb_m)$ is not of type C and also

$$\begin{aligned} \lim _n\lim _m\Omega (a_n,b_m)=\lim _m\lim _n\Omega (a_n,b_m)\not =0. \end{aligned}$$

Thus by Theorem 2.2, $\ell _1(S,\omega )$ is not Arens regular.

As an immediate consequence of Theorem 4.5 we present the next result which has already proved in [1].

Corollary 4.7

For every Brandt semigroup (resp. completely [0-]simple semigroup, [0-]simple inverse semigroup, inverse semigroup with finite set of idempotents) S, the following statements are equivalent.

• (1) $\ell _1(S)$ is Arens regular.

• (2) S is finite.