# Weak amenability of weighted group algebras

M. J. Mehdipour and A. Rejali111Corresponding author

Abstract. In this paper, we study weak amenability of Beurling algebras. To
this end, we introduce the notion inner quasi-additive functions and prove
that for a locally compact group $G$, the Banach algebra $L^{1}(G,\omega)$ is
weakly amenable if and only if every non-inner quasi-additive function in
$L^{\infty}(G,1/\omega)$ is unbounded. This provides an answer to the question
concerning weak amenability of $L^{1}(G,\omega)$ and improve some known
results in connection with it. 11footnotetext:  2020 Mathematics Subject
Classification: 43A20, 47B47,47B48. Keywords: Locally compact group, Weak
amenability, Weighted group algebras.

##  1 Introduction

Throughout this paper $G$ is a locally compact group with an identity element
$e$. Let $L^{1}(G)$ and $M(G)$ be group and measure algebras of $G$,
respectively. Let also $L^{\infty}(G)$ be the Lebesgue space of bounded Borel
measurable functions on $G$; see [4, 8] for an extensive study of these
spaces.

Let us recall that a continuous function $\omega:G\rightarrow[1,\infty)$ is
called a _weight_ _function_ if for every $x,y\in G$

$\omega(xy)\leq\omega(x)\;\omega(y)\quad\hbox{and}\quad\omega(e)=1.$

An elementary computation shows that the functions $\omega^{*}$ and
$\omega^{\otimes}$ defined by

$\omega^{*}(x):=\omega(x)\omega(x^{-1})\quad\hbox{and}\quad\omega^{\otimes}(x,y)=\omega(x)\omega(y)$

are weight functions on $G$ and $G\times G$, respectively. Let the function
$\Omega:G\times G\rightarrow(0,1]$ be defined as follows:

$\Omega(x,y)=\frac{\omega(xy)}{\omega^{\otimes}(x,y)}.$

Then $\Omega$ is called _zero cluster_ if for every pair of sequences
$(x_{n})_{n}$ and $(y_{m})_{m}$ of distinct elements in $G$, we have

$\displaystyle\lim_{n}\lim_{m}\Omega(x_{n},y_{m})=0=\lim_{m}\lim_{n}\Omega(x_{n},y_{m}),$

whenever both iterated limits exist.

Let $L^{1}(G,\omega)$ be the Banach space of all Borel measurable functions
$f$ on $G$ such that $\omega f\in L^{1}(G)$. Then $L^{1}(G,\omega)$ with the
convolution product “$\ast$” and the norm $\|f\|_{\omega}=\|\omega f\|_{1}$ is
a Banach algebras. Let $L^{\infty}(G,1/\omega)$ be the space of all Borel
measurable functions $f$ on $G$ with $f/\omega\in L^{\infty}(G)$. Then
$L^{\infty}(G,1/\omega)$ with the norm

$\|h\|_{\infty,\;\omega}=\|h/\omega\|_{\infty},$

and the multiplication $``\cdot_{\omega}"$ defined by

$h\cdot_{\omega}k=hk/\omega\quad\quad\quad(h,k\in L^{\infty}(G,1/\omega))$

is a commutative $C^{*}-$algebra. Also, $L^{\infty}(G,1/\omega)$ and
$L^{1}(G,\omega)^{*}$ are isometrically isomorphic with the duality given
through

$\langle f,h\rangle=\int_{G}f(x)h(x)\;dx\quad\quad\quad(f\in
L^{1}(G,\omega),h\in L^{\infty}(G,1/\omega)).$

Let $C_{b}(G,1/\omega)$ denote the subspace of $L^{\infty}(G,1/\omega)$
consisting of all bounded continuous functions, and let $C_{0}(G,1/\omega)$ be
the subspace of $C_{b}(G,1/\omega)$ consisting of all functions that vanish at
infinity. Let also $M(G,\omega)$ be the Banach algebra of all complex regular
Borel measures $\mu$ on $G$ for which $\omega\mu\in M(G)$. It is well-known
that $M(G,\omega)$ is the dual space of $C_{0}(G,1/\omega)$ [5, 14, 18], see
[15, 16] for study of weighted semigroup measure algebras; see also [10, 11,
12].

For a Banach algebra $A$, let us recall that a bounded linear operator $D$
from $A$ into $A^{*}$ is called a _derivation_ if $D(ab)=D(a)\cdot b+a\cdot
D(b)$ for all $a,b\in A$. Also, $D$ is said to be an _inner derivation_ if
there exists $z\in A^{*}$ such that for every $a\in A$

$D(a)=\hbox{ad}_{z}(a):=z\cdot a-a\cdot z.$

The space of all continuous (inner) derivations from $A$ into $A^{*}$ is
denoted by (${\cal I}_{nn}(A,A^{*})$) ${\cal Z}(A,A^{*})$, respectively. A
Banach algebra $A$ is called _weakly amenable_ if

${\cal Z}(A,A^{*})={\cal I}_{nn}(A,A^{*}).$

Johnson [9] proved that if $G$ is a locally compact group $G$, then the Banach
algebra $L^{1}(G)$ is weakly amenable; for a simple proof of this result see
[6]. One can arise naturally the question of whether $L^{1}(G,\omega)$ is
weakly amenable. Several authors studied this problem. For example, Bade,
Curtis and Dales [1] characterized weak amenability of the Banach algebra
$L^{1}({{Z}},\omega_{\alpha})$, where $\omega_{\alpha}(n)=(1+|n|)^{\alpha}$.
Gronbaek [7] gave a necessary and sufficient condition for weak amenability of
$L^{1}(G,\omega)$, when $G$ is a discrete Abelian group. He proved that
$\ell^{1}(G,\omega)$ is weakly amenable if and only if there is no non-zero
group homomorphism $\phi:G\rightarrow{{C}}$ such that $\phi\in
C_{b}(G,1/\omega^{*})$. The question comes to mind immediately: Dose this
result hold for any Abelian group? Recently, Zhang [19] gave an affirmative
answer to this conjecture. It is natural to ask whether this result remain
valid for non-Abelian locally compact groups? Borwick [3] studied this
question for non-Abelian discrete groups and gave the conditions that
characterizes weak amenability of $\ell^{1}(G,\omega)$; see also [17] for weak
amenability of $\ell^{1}({{F}}_{2},\omega)$ and
$\ell^{1}(\mathbf{(ax+b)},\omega)$. In this paper, we answer to this
conjecture for non-Abelian locally compact groups.

This paper is organized as follow. In Section 2, we introduce the notions
quasi-additive and inner quasi-additive function. We show that

${\cal Z}(L^{1}(G,\omega),L^{1}(G,\omega)^{*})$

and the space bounded quasi-additive functions are isometrically isomorphic as
Banach spaces. This statement holds for

${\cal I}_{nn}(L^{1}(G,\omega),L^{1}(G,\omega)^{*})$

and the space of inner quasi-additive functions. Using theses result, we prove
that $L^{1}(G,\omega)$ is weakly amenable if and only if every non-inner
quasi-additive function in $L^{\infty}(G,1/\omega)$ is unbounded. This cover
some known results concerning weak amenability of commutative Beurling
algebras. In fact, we give an answer to the question raised in [19]. In
Section 3, we consider two Beurling algebras $L^{1}(G,\omega_{i})$, for
$i=1,2$, and investigate the relation between weak amenability of them. In
section 4, we take up the connection between weak amenability of
$L^{1}(G_{1},\omega_{1})$ and $L^{1}(G_{2},\omega_{2})$.

##  2 Weak amenability of weighted group algebras

Let $G$ be a locally compact group. A Borel measurable function $p:G\times
G\rightarrow{{C}}$ is called _quasi-additive_ if for almost every where
$x,y,z\in G$

$p(xy,z)=p(x,yz)+p(y,zx).$

A quasi-additive function $p$ is called _inner_ if there exists $h\in
L^{\infty}(G,1/\omega)$ such that

$p(x,y)=h(xy)-h(yx)$

for almost every where $x,y\in G$. Let $Q(G)$ be the set of all quasi-additive
functions, $D(G,\omega)$ be the set of all $p\in Q(G)$ such that

$C(p,\omega):=\sup_{x,y\in G}\frac{|p(x,y)|}{\omega^{\otimes}(x,y)}<\infty$

and $I(G,\omega)$ be the set of inner quasi-additive functions. Clearly,

$I(G,\omega)\subseteq D(G,\omega)\subseteq Q(G)\cap L^{\infty}(G\times
G,1/\omega^{\otimes}).$

A quasi-additive function $p$ is called _non-inner in_
$L^{\infty}(G,1/\omega)$ if $p\in Q(G)\setminus I(G,\omega)$. Let
$\check{G}_{\omega}$ be the set of all group homomorphisms
$\phi:G\rightarrow{{C}}$ such that $\phi/\omega$ is bounded. We write
$\check{G}$ for $\check{G}_{1}$. Note that if $p\in D(G,\omega)$ ,
$q\in\check{G}$ and $h\in L^{\infty}(G)$. Then the functions
$p_{1},p_{2},p_{3}:G\times G\rightarrow{{C}}$ defined by $p_{1}(x,y)=q(x)$,

$p_{2}(x,y)=h(xy)-h(yx)\quad\hbox{and}\quad p_{3}(x,y)=p(x^{-1},y^{-1})$

are elements in $D(G,\omega)$. Also, if we define the function
$q_{1}:G\rightarrow{{C}}$ by $q_{1}(x)=p(x,x^{-1})$, then $q_{1}\in\check{G}$.
We now give some properties of quasi-additive functions.

###### Lemma 2.1

Let $G$ be a locally compact group and let $\omega$ and $\omega_{0}$ be weight
functions on $G$. Then the following statements hold.

_(i)_ $D(G,\omega)$ is a closed subspace of $L^{\infty}(G\times
G,1/\omega^{\otimes})$. In particular, $D(G,\omega)$ is a Banach space.

_(ii)_ $I(G,\omega)$ is a subspace of $D(G,\omega)$. Furthermore, if
$C=\\{h\in L^{\infty}(G,1/\omega):h(xy)=h(yx)\;\hbox{for al}l\;x,y\in G\\}$,
then $I(G,\omega)$ and $L^{\infty}(G,1/\omega)/C$ are isomorphic.

_(iii)_ If $\omega_{0}\leq m\omega$ for some $m>0$, then $D(G,\omega_{0})$ and
$I(G,\omega_{0})$ are subspaces of $D(G,\omega)$ and $I(G,\omega)$,
respectively.

_(iv)_ $D(G,\omega)$ and $I(G,\omega)$ are subsets of $D(G,\omega^{*})$ and
$I(G,\omega^{*})$, respectively.

_(v)_ $\check{G}$ can be embedded into $D(G,\omega)$.

Proof. The statement (i) is proved by using standard arguments. For (ii), we
define the function $\Lambda:L^{\infty}(G,1/\omega)\rightarrow I(G,\omega)$ by

$\Lambda(h)(x,y)=h(xy)-h(yx).$

Then $\Lambda$ is an epimorphism with $\hbox{ker}(\Lambda)=C$. Hence
$L^{\infty}(G,1/\omega)/C$ and $I(G,\omega)$ are isomorphic. Hence (ii) holds.
Note that if $\omega_{0}\leq m\omega$ for some $m>0$, then $C(p,\omega)\leq
m^{2}C(p,\omega_{0})$ and $L^{\infty}(G,1/\omega_{0})$ is contained in
$L^{\infty}(G,1/\omega)$. So (iii) holds. The statement (iv) follows from
(iii) and the fact that $\omega\leq\omega^{*}$. Finally, the function
$q\mapsto q\circ\pi_{1}$ from $\check{G}$ onto $D(G,\omega)$ is isomorphism,
where $\pi_{1}$ is the canonical projection. Hence (v) holds.$\hfill\mathchar
1027\relax$

Let $G_{i}$ be a locally compact group and $\omega_{i}$ be a weight function
on $G_{i}$ for $i=1,2$. We define

$\omega_{1}\otimes\omega_{2}(x_{1},x_{2})=\omega_{1}(x_{1})\omega_{2}(x_{2})$

for all $x_{i}\in G_{i}$. It is easy to prove that
$\omega_{1}\otimes\omega_{2}$ is a weight function on $G_{1}\times G_{2}$.

###### Lemma 2.2

Let $G_{i}$ be a locally compact group and let $\omega_{i}$ be a weight
functions on $G_{i}$, for $i=1,2$. Then $D(G_{1}\times
G_{2},\omega_{1}\otimes\omega_{2})$ can be embedded into
$D(G_{1},\omega_{1})\times D(G_{2},\omega_{2})$.

Proof. We only note that the function $p\mapsto(p_{1},p_{2})$ from
$D(G_{1}\times G_{2},\omega_{1}\otimes\omega_{2})$ into
$D(G_{1},\omega_{1})\times D(G_{2},\omega_{2})$ is injective, where

$p_{1}(x_{1},y_{1})=p((x_{1},e_{2}),(y_{1},e_{2}))\quad\hbox{and}\quad
p_{2}(x_{2},y_{2})=p((e_{1},x_{2}),(e_{1},y_{2}))$

for all $x_{1},y_{1}\in G_{1}$ and $x_{2},y_{2}\in G_{2}$.$\hfill\mathchar
1027\relax$

For Banach algebras $A$ and $B$, let $A\hat{\otimes}B$ be the projective
tensor product of $A$ and $B$. Let also ${\cal B}(A,B^{*})$ be the space of
bounded linear operators from $A$ into $B^{*}$. Then the function

$\Gamma:{\cal B}(A,B^{*})\rightarrow(A\hat{\otimes}B)^{*}$

defined by

$\langle\Gamma(T),a\otimes b\rangle=\langle T(a),b\rangle$

is an isometric isomorphism as Banach spaces; see for example Proposition 13
VI in [2]. We now give the next theorem which is actually the key to prove our
results.

###### Theorem 2.3

Let $G$ be a locally compact group and $\omega$ be a weight function on $G$.
Then the following statements hold.

_(i)_ The function $\Gamma:{\cal
Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))\rightarrow D(G,\omega)$ is an
isometric isomorphism, as Banach spaces.

_(ii)_ $\Gamma({\cal
I}_{nn}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))=I(G,\omega)$.

_(iii)_ If $D\in{\cal Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$, then there
exists a unique $p\in D(G,\omega)$ such that $D(f)(x)=\int_{G}p(x,y)f(y)\;dy$
for almost every where $x\in G$.

_(iv)_ If $\omega$ is multiplicative, then $D(G,\omega)=I(G,\omega)$. In
particular, $D(G,1)=I(G,1).$

Proof. (i) Let $D\in{\cal Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. Since
$D$ is a bounded linear operator, setting $A=B=L^{1}(G,\omega)$ in the
definition of $\Gamma$, we have

$p:=\Gamma(D)\in(L^{1}(G,\omega)\hat{\otimes}L^{1}(G,\omega))^{*}=L^{\infty}(G\times
G,1/\omega^{\otimes})$

and

$\displaystyle\langle D(f),g\rangle=\langle p,f\otimes
g\rangle=\int_{G}\int_{G}p(x,y)f(x)g(y)dxdy$ (1)

for all $f,g\in L^{1}(G,\omega)$. On the other hand, for every $f,g,k\in
L^{1}(G,\omega)$ we have

$\displaystyle\langle D(f\ast g),k\rangle$ $\displaystyle=$
$\displaystyle\langle D(f)\cdot g,k\rangle+\langle f\cdot D(g),k\rangle$
$\displaystyle=$ $\displaystyle\langle D(f),g\ast k+k\ast f\rangle.$

It follows that

$\int_{G}\int_{G}p(x,y)(f\ast g)(x)k(y)\;dxdy=\int_{G}\int_{G}p(x,y)f(x)(g\ast
k+k\ast f)(y)\;dxdy$

Thus for every $f,g,h\in L^{1}(G,\omega)$, we obtain

$\displaystyle\int_{G}\int_{G}\int_{G}p(xy,z)f(x)g(y)k(z)\;dxdydz$
$\displaystyle=$
$\displaystyle\int_{G}\int_{G}\int_{G}p(x,yz)f(x)g(y)k(z)\;dxdydz$
$\displaystyle+$
$\displaystyle\int_{G}\int_{G}\int_{G}p(y,zx)f(x)g(y)k(z)\;dxdydz.$

This implies that $p\in D(G,\omega)$. Therefore, $\Gamma$ maps ${\cal
Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$ into $D(G,\omega)$, as Banach
spaces.

Now, let $p\in D(G,\omega)$. Define the linear operator
$D:L^{1}(G,\omega)\rightarrow L^{\infty}(G,1/\omega)$ by

$\langle D(f),g\rangle:=\langle p,f\otimes g\rangle$

for all $f,g\in L^{1}(G,\omega)$. Since $p\in D(G,\omega)$, a similar argument
as given above follows that $D\in{\cal
Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. So $\Gamma(D)=p$. Thus $\Gamma$
is an isometric isomorphism from ${\cal
Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$ onto $D(G,\omega)$.

(ii) First, note that $L^{\infty}(G,1/\omega)$ is a Banach
$L^{1}(G,\omega)-$bimodule with the following actions.

$f\cdot h(x)=\int_{G}f(y)h(xy)\;dy\quad\hbox{and}\quad h\cdot
f(x)=\int_{G}f(y)h(yx)\;dy$

for all $f\in L^{1}(G,\omega)$, $h\in L^{\infty}(G,1/\omega)$ and $x\in G$.

Assume note that $D\in{\cal I}_{nn}(L^{1}(G,\omega),L^{\infty}(G,1/\omega)$.
Then $D=\hbox{ad}_{h}$ for some $h\in L^{\infty}(G,1/\omega)$. So

$\displaystyle\langle D(f),g\rangle$ $\displaystyle=$ $\displaystyle\langle
h\cdot f-f\cdot h,g\rangle$ $\displaystyle=$ $\displaystyle\int_{G}g(y)(h\cdot
f(x)-f\cdot h(x))\;dx$ $\displaystyle=$
$\displaystyle\int_{G}\int_{G}g(y)f(x)(h(xy)-h(yx))\;dxdy$

for all $f,g\in L^{1}(G,\omega)$. Consequently, $\langle D(f),g\rangle=\langle
p,f\otimes g\rangle$, where

$p(x,y)=h(xy)-h(yx)$

for all $x,y\in G$. Obviously, $p\in D(G,\omega)$. This implies that

$\Gamma({\cal I}_{nn}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))=I(G,\omega).$

(iii) Let $D\in{\cal Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. If $f\in
L^{1}(G,\omega)$, then $D(f)\in L^{1}(G,\omega)^{*}$. So for every $g\in
L^{1}(G,\omega)$, we have

$\langle D(f),g\rangle=\int_{G}D(f)(y)g(y)\;dy.$

From this and (1) we conclude that (iii) holds.

(iv) Let $\omega$ be multiplicative. Then $L^{1}(G,\omega)$ and $L^{1}(G)$ are
isomorphism, as Banach algebras. Since $L^{1}(G)$ is weakly amenable, the
statement (v) holds. $\hfill\mathchar 1027\relax$

We now state the main result of this section which answers to an open problem
given in [19].

###### Theorem 2.4

Let $G$ be a locally compact infinite group and $\omega$ be a weight function
on $G$. Then the following assertions are equivalent.

_(a)_ $L^{1}(G,\omega)$ is weakly amenable.

_(b)_ For every bounded derivation $D:L^{1}(G,\omega)\rightarrow
L^{\infty}(G,1/\omega)$ there exists $h\in L^{\infty}(G,1/\omega)$ such that
$\langle D(f),g\rangle=\int_{G}\int_{G}f(x)g(y)(h(xy)-h(yx))\;dxdy$ for all
$f,g\in L^{1}(G,\omega)$.

_(c)_ For every quasi-additive function $p\in D(G,\omega)$ there exists $h\in
L^{\infty}(G,1/\omega)$ such that $p(x,y)=h(xy)-h(yx)$ for all $x,y\in G$.

_(d)_ Every quasi-additive function $p$ with $C(p,\omega)<\infty$ is inner.

_(e)_ Every non-inner quasi-additive function in $L^{\infty}(G,1/\omega)$ is
unbounded.

_(f)_ $D(G,\omega)=I(G,\omega)$.

Proof. Let $D\in{\cal Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. If
$L^{1}(G,\omega)$ is weakly amenable, then $D\in{\cal
I}_{nn}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. By Theorem 2.3 (i),
$\Gamma(D)\in I(G,\omega)$. Thus there exists $h\in L^{\infty}(G,1/\omega)$
such that

$\Gamma(D)(x,y)=h(xy)-h(yx)$

for all $x,y\in G$. It follows from (1) that

$\langle D(f),g\rangle=\int_{G}\int_{G}f(x)g(y)(h(xy)-h(yx))\;dxdy$

for all $f,g\in L^{1}(G,\omega)$. So (a)$\Rightarrow$(b).

Let $p\in D(G,\omega)$. Then $\Gamma(D)=p$ for some $D\in{\cal
Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. If (b) holds, then by (1), for
every $f,g\in L^{1}(G,\omega)$, we have

$\displaystyle\int_{G}\int_{G}p(x,y)f(x)g(y)\;dxdy$ $\displaystyle=$
$\displaystyle\langle\Gamma(D),f\otimes g\rangle=\langle D(f),g\rangle$
$\displaystyle=$ $\displaystyle\int_{G}\int_{G}f(x)g(y)(h(xy)-h(yx))\;dxdy.$

This shows that $p(x,y)=h(xy)-h(yx)$ for almost every where $x,y\in G$. So
(b)$\Rightarrow$(c). For the implication (c)$\Rightarrow$(d), note that

$D(G,\omega)=\\{p\in Q(G):C(p,\omega)<\infty\\}.$

The implications (d)$\Rightarrow$(e)$\Rightarrow$(f) are clear. Finally, let
$D\in{\cal Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. Then $\Gamma(D)=p$ for
some $p\in D(G,\omega)$. If $p\in I(G,\omega)$, then $D\in{\cal
I}_{nn}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))$. That is, (d) implies (a).
$\hfill\mathchar 1027\relax$

As a consequence of Theorem 2.4 we give the following result.

###### Corollary 2.5

Let $G$ be a locally compact infinite group. Then $L^{1}(G,\omega)$ is not
weakly amenable if and only if there exists a bounded non-inner quasi-additive
function in $L^{\infty}(G,1/\omega)$.

The next result can be considered as an improvement of Theorem 3.1 in [19].

###### Corollary 2.6

Let $G$ be a locally compact Abelian group. Then the following assertions are
equivalent.

_(a)_ $L^{1}(G,\omega)$ is weakly amenable.

_(b)_ The zero map is the only quasi-additive function in $D(G,\omega)$.

_(c)_ The zero map is the only group homomorphism in
$L^{\infty}(G,1/\omega^{*})$.

_(d)_ The zero map is the only group homomorphism in $C_{b}(G,1/\omega^{*})$.

Proof. From Theorem 1.2 in [19] and Theorem 2.4, the implications
(d)$\Rightarrow$(a)$\Rightarrow$(b) hold. The implication (c)$\Rightarrow$(d)
is clear. To show that (b)$\Rightarrow$(c), let $\phi$ be a group homomorphism
in $L^{\infty}(G,1/\omega^{*})$. Define $p:G\times G\rightarrow{{C}}$ by
$p(x,y)=\phi(x)$. Then $p$ is a quasi-additive function in $L^{\infty}(G\times
G,1/\omega^{\otimes})$. So $\phi=0$.$\hfill\mathchar 1027\relax$

In the sequel, we give another consequent of Theorem 2.4.

###### Corollary 2.7

Let $G$ be a locally compact infinite group. Then the following statements
hold.

_(i)_ Every non-inner quasi-additive function in $L^{\infty}(G)$ is unbounded.

_(ii)_ If $G$ is Abelian, then every non-zero group homomorphism from $G$ into
${{C}}$ is unbounded.

Proof. It is well-known that if $G$ is a locally compact group, then
$L^{1}(G)$ is weakly amenable; see for example [5]. This together with Theorem
2.4 establish (i). The statement (ii) follows from Corollary
2.6.$\hfill\mathchar 1027\relax$

###### Example 2.8

Define the function $\omega:{R}\rightarrow[1,\infty)$ by

$\displaystyle\omega(x)=\left\\{\begin{array}[]{rl}\hbox{exp}(x)&x\geq 0\\\
1&x<0\end{array}\right.$

Then $\omega$ is a weight function on ${R}$. For $r\in{R}$, we define
$p_{r}(x)=rx$ for all $x\in{R}$. Then

$\sup{\frac{|p_{r}(x)|}{\omega^{*}(x)}:x\in{R}\\}=\sup\\{|rx|}{\hbox{exp}(x)}:x\in{R}^{+}\\}\leq|r|.$

Hence there exists a non-zero group homomorphism in
$L^{\infty}({R},1/\omega^{*})$. Thus $L^{1}({R},\omega)$ is not weakly
amenable. It is proved that

$D(L^{1}(G,\omega))=[-1,0]\times{R}.$

##  3 Beurling algebras with different weights

We commence this section with the following result which is interesting and
useful.

###### Theorem 3.1

Let $\omega_{1}$ and $\omega_{2}$ be weight functions on a locally compact
group $G$. Then the following assertions are equivalent.

_(a)_ $L^{1}(G,\omega_{2})$ is a subspace of $L^{1}(G,\omega_{1})$.

_(b)_ $M(G,\omega_{2})$ is a subspace of $M(G,\omega_{1})$.

_(c)_ There exists $m>0$ such that $\|f\|_{\omega_{1}}\leq
m\|f\|_{\omega_{2}}$ for all $f\in L^{1}(G,\omega_{2})$.

_(d)_ There exists $n>0$ such that $\|\mu\|_{\omega_{2}}\leq
n\|\mu\|_{\omega_{1}}$ for all $\mu\in M(G,\omega_{2})$.

_(e)_ There exists $s>0$ such that $\omega_{1}\leq s\omega_{2}$.

Proof. The implications (e)$\Rightarrow$(c)$\Rightarrow$(a) are clear. If
$L^{1}(G,\omega_{2})$ is a subspace of $L^{1}(G,\omega_{1})$, we can regard
$L^{1}(G,\omega_{2})^{**}$ as a subspace of $L^{1}(G,\omega_{1})^{**}$. Hence
$M(G,\omega_{1})\oplus C_{0}(G,1/\omega_{1})^{\perp}$ is contained in
$M(G,\omega_{2})\oplus C_{0}(G,1/\omega_{2})^{\perp}$. This shows that
$M(G,\omega_{1})=\pi_{1}(L^{1}(G,\omega_{1})^{**})$ is a subset of
$M(G,\omega_{2})=\pi_{1}(L^{1}(G,\omega_{2})^{**})$. That is,
(a)$\Rightarrow$(b). To show that (b)$\Rightarrow$(d), we define

$\|\mu\|=\|\mu\|_{\omega_{1}}+\|\mu\|_{\omega_{2}},$

then $B:=(M(G,\omega_{2}),\|.\|)$ is a Banach algebra. So the identity map
$F:B\rightarrow A$ is injective and continuous, where
$A:=(M(G,\omega_{1}),\|.\|_{\omega_{1}})$. So $F:B\rightarrow F(B)$ is
bijective. In view of the inverse mapping theorem, $F^{-1}$ is continuous.
Thus there exists $r>1$ such that

$\|F^{-1}(\mu)\|\leq r\|\mu\|_{\omega_{1}}$

for all $\mu\in M(G,\omega_{1})$. Consequently,

$\|\mu\|_{\omega_{1}}+\|\mu\|_{\omega_{2}}\leq r\|\mu\|_{\omega_{1}}.$

Therefore, $\|\mu\|_{\omega_{2}}\leq(r-1)\|\mu\|_{\omega_{1}}$. Finally, let
(d) hold. Then for every $x\in G$, we have $\|\delta_{x}\|_{\omega_{2}}\leq
n\|\delta_{x}\|_{\omega_{1}}$. Thus $\omega_{1}\leq n\omega_{2}$ for some
$n>0$. That is, (d)$\Rightarrow$(e).$\hfill\mathchar 1027\relax$

Let us recall that two weight function $\omega_{1}$ and $\omega_{2}$ are
_equivalent_ if $m\omega_{1}\leq\omega_{2}\leq n\omega_{1}$ for some $m,n>0$.

###### Corollary 3.2

Let $\omega_{1}$ and $\omega_{2}$ be weight functions on a locally compact
group $G$. Then the following assertions are equivalent.

_(a)_ $L^{1}(G,\omega_{1})=L^{1}(G,\omega_{2})$.

_(b)_ $\|.\|_{\omega_{1}}$ and $\|.\|_{\omega_{2}}$ are equivalent.

_(c)_ $\omega_{1}$ and $\omega_{2}$ are equivalent.

In the following, we consider two weight functions $\omega_{1}$ and
$\omega_{2}$ on a locally compact group $G$ and study the relation between
weak amenability of $L^{1}(G,\omega_{1})$ and $L^{1}(G,\omega_{2})$.

###### Proposition 3.3

Let $\omega_{1}$ and $\omega_{2}$ be weight functions on a locally compact
group $G$ such that $\omega_{1}\leq m\omega_{2}$ for some $m>0$. Then the
following statements hold.

_(i)_ If $L^{1}(G,\omega_{2})$ is weakly amenable and
$I(G,\omega_{1})=I(G,\omega_{2})$, then $L^{1}(G,\omega_{1})$ is weakly
amenable.

_(ii)_ If $\omega_{2}$ or $\omega_{2}^{*}$ is bounded, then
$L^{1}(G,\omega_{1})$ is weakly amenable.

Proof. (i) By Lemma 2.1 and Theorem 2.4 we have

$D(G,\omega_{1})\subseteq
D(G,\omega_{2})=I(G,\omega_{2})=I(G,\omega_{1})\subseteq D(G,\omega_{1}).$

Hence $D(G,\omega_{1})=I(G,\omega_{1})$ and so $L^{1}(G,\omega_{1})$ is weakly
amenable.

(ii) Let $\omega_{2}$ be bounded. Then $\omega_{1}$ is bounded and so
$L^{1}(G,\omega_{1})$ is weakly amenable. Now if $\omega_{2}^{*}$ is bounded,
then $\omega_{2}$ is bounded, because $\omega_{2}\leq\omega_{2}^{*}$. Hence
$L^{1}(G,\omega_{1})$ is weakly amenable.$\hfill\mathchar 1027\relax$

As an immediate corollary of Proposition 3.3 we have the following result.

###### Corollary 3.4

Let $\omega_{1}$ and $\omega_{2}$ be weight functions on a locally compact
group $G$ such that $\omega_{1}$ and $\omega_{2}$ are equivalent. Then
$L^{1}(G,\omega_{1})$ is weakly amenable if and only if $L^{1}(G,\omega_{2})$
is weakly amenable.

###### Example 3.5

Let $\omega$ be a weight function on a locally compact group $G$ and $a\in G$.
The function $\omega_{a}$ defined by $\omega_{a}(x)=\omega(axa^{-1})$ is a
weight function on $G$. For every $x\in G$, we have

$\displaystyle\omega(x)=\omega(a^{-1}(axa^{-1})a)\leq\omega(a^{-1})\omega_{a}(x)\omega(a)$

and

$\omega_{a}(x)=\omega(axa^{-1})\leq\omega(a)\omega(a^{-1})\omega(x).$

Hence

$\displaystyle(1/m)\omega(x)\leq\omega_{a}(x)\leq m\omega(x),$ (4)

where $m=\omega(a)\omega(a^{-1})$. That is, $\omega$ and $\omega_{a}$ are
equivalent. Now, define the function $W$ on $G$ by

$W(x)=\inf_{a\in G}\omega_{a}(x).$

Then $W$ is a weight function on $G$. By (4), $\omega$ and $W$ are equivalent.
Hence $L^{1}(G,\omega)$ is weakly amenable if and only if
$L^{1}(G,\omega_{a})$ is weakly amenable; or equivalently, $L^{1}(G,W)$ is
weakly amenable.

As another consequence of Theorem 3.6 we have the following result. The part
(i) due to Pourabbas [13].

###### Corollary 3.6

Let $G$ be a locally compact group. Then the following statements hold.

_(i)_ If $\omega^{*}$ is bounded, then $L^{1}(G,\omega)$ is weakly amenable
and $D(G,\omega^{*})=I(G,\omega^{*})$.

_(ii)_ If $L^{1}(G,\omega^{*})$ is weakly amenable and
$I(G,\omega)=I(G,\omega^{*})$, then $L^{1}(G,\omega)$ is weakly amenable and
$D(G,\omega)=D(G,\omega^{*})$. In the case where $G$ is Abelian,
$D(G,\omega)=\\{0\\}$.

_(iii)_ If $D(G,\omega^{*})=I(G,\omega^{*})=I(G,\omega)$, then
$D(G,\omega)=I(G,\omega)$.

Proof. Since $\omega\leq\omega^{*}$, the statement (i) follows from
Proposition 3.3 (ii). From Proposition 3.3 yields that $L^{1}(G,\omega)$ is
weakly amenable and by Theorem 2.4,

$D(G,\omega)=I(G,\omega)=I(G,\omega^{*})=D(G,\omega^{*}).$

When $G$ is Abelian, we have

${\cal Z}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))={\cal
I}_{nn}(L^{1}(G,\omega),L^{\infty}(G,1/\omega))=\\{0\\}.$

In view of Theorem 2.3, we infer that $D(G,\omega)=I(G,\omega)=\\{0\\}$. So
(ii) holds. For (iii), by Lemma 2.1 we have

$D(G,\omega)\subseteq D(G,\omega^{*})=I(G,\omega^{*})=I(G,\omega)\subseteq
D(G,\omega).$

Hence (iii) holds.$\hfill\mathchar 1027\relax$

###### Example 3.7

It is well-known from [1] that $\ell^{1}({{Z}},\omega_{\alpha})$ is weakly
amenable if and only if $0\leq\alpha<1/2$. Hence the Banach algebra
$\ell^{1}({{Z}},\omega_{1/3})$ is weakly amenable, however,
$\ell^{1}({{Z}},\omega_{1/3}^{*})$ is not weakly amenable. Also,
$\ell^{1}({{Z}},\omega_{1/6}^{*})$ is weakly amenable, but,
$\omega_{\alpha}^{*}$ is unbounded.

Let $\omega$ be a weight function on $G$. Then $\omega^{\prime}$ defined by
$\omega^{\prime}(x)=\omega(x^{-1})$ for all $x\in G$ is also a weight function
on $G$.

###### Theorem 3.8

Let $G$ be a locally compact group and $\omega$ be a weight function on $G$.
Then $L^{1}(G,\omega^{\prime})$ is weakly amenable if and only if
$L^{1}(G,\omega)$ is weakly amenable.

Proof. Let $p$ be a complex-valued function $G\times G$. Define
$p^{\prime}(x,y)=p(x^{-1},y^{-1})$ for all $x,y\in G$. It routine to check
that

$C(p,\omega)=C(p^{\prime},\omega^{\prime}).$

Now, for a complex-valued function $h$ on $G$, define
$h^{\prime}(x)=-h(x^{-1})$ for all $x\in G$. Obviously, $p(x,y)=h(xy)-h(yx)$
if and only if $h^{\prime}(x,y)=h^{\prime}(xy)-h^{\prime}(yx)$ for all $x,y\in
G$. Therefore, $p\in I(G,\omega)$ if and only if $p^{\prime}\in
I(G,\omega^{\prime})$. These facts together with Theorem 2.4 prove the
result.$\hfill\mathchar 1027\relax$

It is easy to see that if $G$ is a locally compact Abelian group, then
${D}:=\\{(x,x^{-1}):x\in G\\}$ is a locally compact Abelian group.

###### Theorem 3.9

Let $G$ be a locally compact Abelian group and $\omega$ be a weight function
on $G$. Then $L^{1}(G,\omega^{*})$ is weakly amenable if and only if
$L^{1}({D},\omega^{\otimes})$ is weakly amenable.

_(ii)_ $L^{1}(G,\omega^{\prime})$ is weakly amenable if and only if
$L^{1}(G,\omega)$ is weakly amenable, where
$\omega^{\prime}(x)=\omega(x^{-1})$ for all $x\in G$.

Proof. Let $q:G\rightarrow{{C}}$ be a group homomorphism. Define the function
$Q:{D}\rightarrow{{C}}$ by $Q(x,x^{-1})=q(x)$. Then $Q$ is a group
homomorphism. Since for every $x\in G$

$\frac{|q(x)|}{(\omega^{*})^{*}(x)}=\frac{|Q(x,x^{-1})|}{(\omega^{\otimes})^{*}(x,x^{-1})},$

it follows that $q\in\check{G}_{(\omega^{*})^{*}}$ if and only if
$Q\in\check{{D}}_{(\omega^{\otimes})^{*}}$. Also, $q=0$ if and only if $Q=0$.
Now, apply Corollary 2.6. $\hfill\mathchar 1027\relax$

##  4 Weak amenability of Beurling algebras

Let $\phi:G_{1}\rightarrow G_{2}$ be a group epimorphism and $\omega$ be a
weight function on $G_{2}$. Then the function
$\overleftarrow{\omega}:G_{1}\rightarrow[1,\infty)$ defined by
$\overleftarrow{\omega}(x_{1})=\omega(\phi(x_{1}))$ is a weight function on
$G_{1}$. Define the function ${S}:Q(G_{2})\rightarrow Q(G_{1})$ by

${S}(p_{2})(x_{1},y_{1})=p_{2}(\phi(x_{1}),\phi(y_{1})).$

It is clear that ${S}$ is injective and
$C(p_{2},\omega)=C({S}(p_{2}),\overleftarrow{\omega})$ for all $p_{2}\in
Q(G_{2})$. So ${S}$ maps $D(G_{2},\omega)$ into
$D(G_{1},\overleftarrow{\omega})$. Since $h_{2}\circ\phi\in
L^{\infty}(G_{1},1/\overleftarrow{\omega})$ for all $h_{2}\in
L^{\infty}(G_{2},1/\omega)$. This shows that ${S}(I(G_{2},\omega))$ is
contained in $I(G_{1},\overleftarrow{\omega})$.

###### Proposition 4.1

Let $G_{1}$ and $G_{2}$ be locally compact infinite groups and let $\omega$ be
a weight function on $G_{2}$. Then the following statements hold.

_(i)_ If $L^{1}(G_{2},\omega)$ is weakly amenable, then
$L^{1}(G_{1},\overleftarrow{\omega})$ is weakly amenable.

_(ii)_ $L^{1}(G_{1},\overleftarrow{\omega})$ is weakly amenable and
${S}(I(G_{2},\omega))=I(G_{1},\overleftarrow{\omega})$, then
$L^{1}(G_{2},\omega)$ is weakly amenable.

Proof. (i) Let $p_{1}$ be a non-inner quasi-additive function in
$L^{\infty}(G_{1},1/\overleftarrow{\omega})$. Since ${S}$ maps
$I(G_{2},\omega)$ into $I(G_{1},\overleftarrow{\omega})$, there exists a non-
inner quasi-additive function $p_{2}$ in $L^{\infty}(G_{2},1/\omega)$ such
that ${S}(p_{2})=p_{1}$. So if $L^{1}(G_{2},\omega)$ is weakly amenable, then

$C(p_{1},\overleftarrow{\omega})=C({S}(p_{2}),\overleftarrow{\omega})=C(p_{2},\omega)=\infty.$

Therefore, $L^{1}(G_{1},\overleftarrow{\omega})$ is weakly amenable.

(ii) Let $p_{2}$ be a non-inner quasi-additive function in
$L^{\infty}(G_{2},1/\omega)$. Since ${S}$ is injective, ${S}(p_{2})$ is a non-
inner quasi-additive function in $L^{\infty}(G_{1},1/\overleftarrow{\omega})$.
Hence if $L^{1}(G_{1},\overleftarrow{\omega})$ is weakly amenable, then

$C(p_{2},\omega)=C({S}(p_{2}),\overleftarrow{\omega})=\infty.$

This shows that $L^{1}(G_{2},\omega)$ is weakly amenable.$\hfill\mathchar
1027\relax$

Let $G$ be a locally compact group and $N$ be a normal subgroup of $G$. It is
easy to see that the function

$\hat{\omega}(xN)=\inf\\{\omega(y):yN=xN\\}$

is a weight function on $G/N$. Also the function
$\overline{\omega}(x)=\hat{\omega}(xN)$ is a weight function on $G$ and
$\overline{\omega}(x)\leq\omega(x)$ for all $x\in G$.

###### Corollary 4.2

Let $\omega$ be a weight function on a locally compact group $G$ and $N$ be a
normal subgroup of $G$ such that $G/N$ is infinite. If
$L^{1}(G/N,\hat{\omega})$ is weakly amenable, then
$L^{1}(G,\overline{\omega})$ is weakly amenable.

Proof. Let $\pi:G\rightarrow G/N$ be the quotient map. Then for every $x\in
G$, we have

$\overleftarrow{\omega}(x)=\hat{\omega}(\pi(x))=\hat{\omega}(xN)=\overline{\omega}(x).$

Hence $\overleftarrow{\omega}=\overline{\omega}$. Now, invoke Proposition
4.1.$\hfill\mathchar 1027\relax$

Let $\omega_{i}$ be a weight function on a locally compact group $G_{i}$, for
$i=1,2$. Then $W_{i}(x_{1},x_{2})=\omega_{i}(x_{i})$ is a weight function on
$G_{1}\times G_{2}$, for $i=1,2$.

###### Corollary 4.3

Let $\omega_{i}$ be a weight function on a locally compact infinite group
$G_{i}$, for $i=1,2$. If $L^{1}(G_{1},\omega_{1})$ or
$L^{1}(G_{2},\omega_{2})$ is weakly amenable, then $L^{1}(G_{1}\times
G_{2},W_{i})$ is weakly amenable.

Proof. Let $\pi_{i}:G_{1}\times G_{2}\rightarrow G_{i}$ be the canonical
projection, for $i=1,2$. Then

$\overleftarrow{\omega}(x_{1},x_{2})=\omega_{i}(\pi_{i}(x_{1},x_{2}))=\omega_{i}(x_{i})=W_{i}(x_{1},x_{2})$

for all $x_{i}\in G_{i}$. So $\overleftarrow{\omega}=W_{i}$. By Proposition
4.1, the result hold.$\hfill\mathchar 1027\relax$

In the following we investigate the relation between weak amenability of
$L^{1}(G,\omega)$ and $L^{1}(H,\omega|_{H})$ when $H$ is a subgroup of $G$. To
this end, we need some results.

###### Proposition 4.4

Let $\omega$ be a weight function on a locally compact Abelian group $G$ and
$H$ be a subgroup of $G$. Then every group homomorphism $q:H\rightarrow{{C}}$
has an extension to a group homomorphism $Q:G\rightarrow{{C}}$.

Proof. Let $q:H\rightarrow{{C}}$ be a group homomorphism. Let $A$ be a subset
of $G$ such that $\\{Ha:a\in A\\}$ is a family of pairwise disjoint subsets of
$G$ and $G=\cup_{a\in A}Ha$. If $x\in G$, then there exists a unique element
$a_{x}\in A$ such that $x\in Ha_{x}$. So there exists a unique element
$\tilde{x}=x(a_{x})^{-1}$ in $H$ such that $x=\tilde{x}a_{x}$. Define
$Q(x)=q(\tilde{x})$. For every $x,y\in G$, we have $Ha_{x}Ha_{y}=Ha_{x}a_{y}$.
Thus $xy\in Ha_{x}a_{y}\cap Ha_{xy}$. Hence $Ha_{x}a_{y}=Ha_{xy}$. It follows
that $a_{x}a_{y}=a_{xy}$. Hence

$\tilde{x}\tilde{y}a_{x}a_{y}=\tilde{x}a_{x}\tilde{y}a_{y}=xy=\mathaccent
869{xy}a_{xy}=\mathaccent 869{xy}a_{x}a_{y}.$

and thus $\mathaccent 869{xy}=\tilde{x}\tilde{y}$. Consequently,

$Q(xy)=q(\mathaccent 869{xy})=q(\tilde{x})+q(\tilde{y})=Q(x)+Q(y).$

This shows that $Q$ is a group homomorphism. For every $h\in H$, we have
$he=h$. Thus $a_{x}=e$. Therefore, $x=\tilde{x}$. That is,
$Q|_{H}=q$.$\hfill\mathchar 1027\relax$

Let $G$ be a locally compact Abelian group and $H$ be a subgroup of $G$. We
denote by $\mathaccent 869{H}$ be the set of all $\tilde{x}$ pointed out in
the proof of Proposition 4.4.

###### Lemma 4.5

Let $G$ be a locally compact Abelian group and $\omega$ be a weight function
on $G$. Then the following statements hold.

_(i)_ $\mathaccent 869{H}$ is a normal subgroup of $H$.

_(ii)_ $\omega^{*}(\tilde{x})=\omega\otimes\omega(\tilde{x},(\tilde{x})^{-1})$
for all $x\in G$.

_(iii)_ The function $\tilde{\omega}$ defined by
$\tilde{\omega}(x)=\omega(\tilde{x})$ is a weight function on $G$.
Furthermore, $(\tilde{\omega})^{*}(x)=\omega^{*}(\tilde{x})$ for all $x\in G$.

Proof. It is easy to see that $\tilde{e}=e$ and $\mathaccent
869{xy}=\tilde{x}\tilde{y}$ for all $x,y\in G$. since $xx^{-1}=e$, we have
$\tilde{x}\mathaccent 869{x^{-1}}=\tilde{e}=e$. Thus $\mathaccent
869{x^{-1}}=(\tilde{x})^{-1}$. These facts show that $\mathaccent 869{H}$ is a
subgroup of $G$. So (i) holds. It is easy to prove that (ii) holds. Since
$\mathaccent 869{xy}=\tilde{x}\tilde{y}$, it follows that

$\displaystyle\tilde{\omega}(xy)$ $\displaystyle=$
$\displaystyle\omega(\mathaccent 869{xy})=\omega(\tilde{x}\tilde{y})$
$\displaystyle\leq$
$\displaystyle\omega(\tilde{x})\omega(\tilde{y})=\tilde{\omega}(x)\tilde{\omega}(y).$

This shows that $\tilde{\omega}$ is a weight function on $G$. For every $x\in
G$, we have

$\displaystyle(\tilde{\omega})^{*}(x)$ $\displaystyle=$
$\displaystyle\tilde{\omega}(x)\tilde{\omega}(x^{-1})$ $\displaystyle=$
$\displaystyle\omega(\tilde{x})\omega(\mathaccent 869{x^{-1}})$
$\displaystyle=$ $\displaystyle\omega(\tilde{x})\omega((\tilde{x})^{-1})$
$\displaystyle=$ $\displaystyle\omega^{*}(\tilde{x}).$

Therefore, $(\tilde{\omega})^{*}(x)=\omega^{*}(\tilde{x})$ for all $x\in G$.
That is, (iii) holds. $\hfill\mathchar 1027\relax$

###### Theorem 4.6

Let $G$ be a locally compact Abelian group and $H$ be a subgroup of $G$. If
$L^{1}(G,\tilde{\omega})$ is weakly amenable, then $L^{1}(H,\omega|_{H})$ is
weakly amenable.

Proof. Let $q:H\rightarrow{C}$ be a non-zero group homomorphism. By
Proposition 4.4, there exists $Q:G\rightarrow{C}$ such that
$Q(x)=q(\tilde{x})$ for all $x\in G$. Hence

$\displaystyle\sup\\{\frac{|q(y)|}{\omega^{*}(y)}:y\in H\\}$
$\displaystyle\geq$
$\displaystyle\sup\\{\frac{|q(\tilde{x})|}{\omega^{*}(\tilde{x})}:x\in G\\}$
$\displaystyle=$
$\displaystyle\sup\\{\frac{|Q(x)|}{\omega^{*}(\tilde{x})}:x\in
G\\}=\sup\\{\frac{|Q(x)|}{(\tilde{\omega})^{*}(x)}:x\in G\\}.$

Therefore, $L^{1}(H,\omega|_{H})$ is weakly amenable.$\hfill\mathchar
1027\relax$

Now, we investigate weak amenability of tensor product of Beurling algebras.

###### Theorem 4.7

Let $G_{1}$ and $G_{2}$ be locally compact groups and let $\omega_{1}$ and
$\omega$ be weight functions on $G_{1}$ and $G_{1}\times G_{2}$, respectively.
If $L^{1}(G_{1}\times G_{2},\omega)$ is weakly amenable and
$\sup\\{\frac{\omega_{1}(x)}{\omega(x,y)}:x,y\in G\\}<\infty$, then
$L^{1}(G_{1},\omega_{1})$ is weakly amenable.

Proof. Let $p$ be a non-inner quasi-additive function in
$L^{\infty}(G_{1},1/\omega_{1})$. Define the function
$\tilde{p}((x_{1},y_{1}),(x_{2},y_{2}))=p(x_{1},x_{2})$ for all
$x_{1},y_{1}\in G_{1}$ and $x_{2},y_{2}\in G_{2}$. Suppose that $\tilde{p}\in
I(G_{1}\times G_{2},\omega_{1}\otimes\omega_{2})$. Then there exists $h\in
L^{\infty}(G_{1}\times G_{2},\omega_{1}\times\omega_{2})$ such that

$\tilde{p}(x_{1},x_{2}),(y_{1},y_{2}))=h((x_{1},y_{1})(x_{2},y_{2}))-h((x_{2},y_{2})(x_{1},y_{1}))$

for all $x_{1},y_{1}\in G_{1}$ and $x_{2},y_{2}\in G_{2}$. Define the complex-
valued function $k$ on $G_{1}$ by $k(x)=h(x,e_{2})$ for all $x\in G_{1}$. Note
that

$\frac{k(x_{1})}{\omega_{1}(x_{1})}=\frac{h(x_{1},e_{2})}{\omega_{1}\otimes\omega_{2}(x_{1},e_{2})}.$

This shows that $k\in L^{\infty}(G_{1},1/\omega_{1})$. For every
$x_{1},y_{1}\in G_{1}$ and $x_{2},y_{2}\in G_{2}$, we have

$\displaystyle k(x_{1}y_{1})-k(y_{1}x_{1})$ $\displaystyle=$ $\displaystyle
h((x_{1},e_{2})(y_{1},e_{2}))-h((y_{1},e_{2})(x_{1},e_{2}))$ $\displaystyle=$
$\displaystyle\tilde{p}((x_{1},e_{2}),(y_{1},e_{2}))$ $\displaystyle=$
$\displaystyle p(x_{1},y_{1}).$

This contradiction shows that $\tilde{p}$ is non-inner. If $x_{1},y_{1}\in
G_{1}$ and $x_{2},y_{2}\in G_{2}$,then

$\frac{|\tilde{p}(x_{1},x_{2}),(y_{1},y_{2}))|}{\omega(x_{1},x_{2})\omega(y_{1},y_{2})}=\frac{|p(x_{1},y_{1})|}{\omega_{1}(x_{1})\omega_{1}(y_{1})}\frac{\omega_{1}(x_{1})}{\omega(x_{1},x_{2})}\frac{\omega_{1}(y_{1})}{\omega(y_{1},y_{2})}.$

This implies that if $L^{1}(G_{1},\omega_{1})$ is not weakly amenable, then
$L^{1}(G_{1}\times G_{2},\omega)$ is not weakly amenable.$\hfill\mathchar
1027\relax$

For a locally compact Abelian group $G$, Zhang [19] proved that if
$L^{1}(G_{1},\omega_{1})\hat{\otimes}L^{1}(G_{2},\omega_{2})$ is weakly
amenable, then $L^{1}(G_{1},\omega_{1})$ and $L^{1}(G_{2},\omega_{2})$ are
weakly amenable. We show that this result is true for any locally compact
group.

###### Corollary 4.8

Let $\omega_{i}$ be a weight function on a locally compact infinite group
$G_{i}$, for $i=1,2$. If
$L^{1}(G_{1},\omega_{1})\hat{\otimes}L^{1}(G_{2},\omega_{2})$ is weakly
amenable, then $L^{1}(G_{1},\omega_{1})$ and $L^{1}(G_{2},\omega_{2})$ are
weakly amenable.

Proof. Let $L^{1}(G_{1},\omega_{1})\hat{\otimes}L^{1}(G_{2},\omega_{2})$ be
weakly amenable. Then $L^{1}(G_{1}\times G_{2},\omega_{1}\otimes\omega_{2})$
is weakly amenable. Since for every $x,y\in G$

$\frac{\omega_{1}(x)}{\omega_{1}\otimes\omega_{2}(x,y)}=\frac{1}{\omega_{2}(y)}\leq
1,$

it follows from Theorems 4.7 that $L^{1}(G_{1},\omega_{1})$ is weakly
amenable. The other case is similar.$\hfill\mathchar 1027\relax$

## References

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Mohammad Javad Mehdipour
Department of Mathematics,
Shiraz University of Technology,
Shiraz 71555-313, Iran
e-mail<EMAIL_ADDRESS>
Ali Rejali
Department of Pure Mathematics,
Faculty of Mathematics and Statistics,
University of Isfahan,
Isfahan 81746-73441, Iran
e-mail<EMAIL_ADDRESS>


# Low-Metallicity Globular Clusters in the Low-Mass Isolated Spiral Galaxy NGC
2403

Duncan A. Forbes,1 Anna Ferré-Mateu,2,1 Jonah S. Gannon1, Aaron J.
Romanowsky3,4,5, Jeffrey L. Carlin6, Jean P. Brodie1,4 and Jacob Day3
1Centre for Astrophysics & Supercomputing, Swinburne University of Technology,
Hawthorn VIC 3122, Australia
2Instituto Astrofisica de Canarias, Av. Via Lactea s/n, E38205 La Laguna,
Spain
3Department of Physics and Astronomy, San José State University, San José, CA
95192, USA
4University of California Observatories, 1156 High St., Santa Cruz, CA 95064,
USA
5Department of Astronomy and Astrophysics, University of California, Santa
Cruz, CA 95064, USA
6NSF’s NOIRLab/Rubin Observatory Project Office, 950 North Cherry Avenue,
Tucson, AZ 85719, USA E-mail<EMAIL_ADDRESS>(DAF)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The globular cluster (GC) systems of low-mass late-type galaxies, such as NGC
2403, have been poorly studied to date. As a low mass galaxy (M∗ = 7 $\times$
109 M⊙), cosmological simulations predict NGC 2403 to contain few, if any,
accreted GCs. It is also isolated, with a remarkably undisturbed HI disk.
Based on candidates from the literature, Sloan Digital Sky Survey and Hyper
Suprime-Cam imaging, we selected several GCs for follow-up spectroscopy using
the Keck Cosmic Web Imager. From their radial velocities, and other
properties, we identify 8 bona-fide GCs associated with either the inner halo
or the disk of this bulgeless galaxy. A stellar population analysis suggests a
wide range of GC ages from shortly after the Big Bang until the present day.
We find all of the old GCs to be metal-poor with [Fe/H] $\leq$ –1. The
age–metallicity relation for the observed GCs suggests that they were formed
over many Gyr from gas with a low effective yield, similar to that observed in
the SMC. Outflows of enriched material may have contributed to the low yield.
With a total system of $\sim$50 GCs expected, our study is the first step in
fully mapping the star cluster history of NGC 2403 in both space and time.

###### keywords:

galaxies: individual (NGC 2403)– galaxies: star clusters – galaxies: evolution

††pubyear: 2022††pagerange: Low-Metallicity Globular Clusters in the Low-Mass
Isolated Spiral Galaxy NGC 2403–A

##  1 Introduction

The globular cluster (GC) systems of galaxies is an area of active research
(see review by Forbes et al. (2018) and references therein). This is
particularly true for low-mass galaxies, which have relatively poor GC systems
compared to giant galaxies with their populous GC systems. While the Local
Group contains some well-known low mass galaxies with modest GC systems, such
as M33, the LMC and the SMC, the study of their GC systems is in some ways
complicated by their proximity. In the case of M33 its radial velocities of
its GCs ($\sim$ -180 km/s) overlap with high velocity stars of the Milky Way;
the recent work of Larsen et al. (2021) found that several claimed GCs, in
previous studies of M33, are actually foreground Milky Way stars. The
Magellanic Clouds are of course in the process of merging with the Milky Way
and it has been speculated for some time that they may have ‘lost’ GCs to our
Galaxy (Lynden-Bell & Lynden-Bell, 1995). Thus we need to look further afield
to improve our understanding of the GC systems of such low mass galaxies.

While GC systems, in general, include both in-situ and ex-situ (accreted)
formed GCs (Forbes & Remus, 2018), low mass galaxies are expected to be
dominated by in-situ formed GCs e.g. Oser et al. (2010) and Remus & Forbes
(2021). In particular, galaxies with a total halo mass of Mh $<$ 1011 M⊙ are
expected to be dominated by in-situ formed GCs (Choksi & Gnedin, 2019). This
is in contrast to massive galaxies like the Milky Way, the halos of which are
dominated by the accreted GCs of disrupted dwarf galaxies (Forbes, 2020).

NGC 2403 is a low mass (MV = –19.5, M∗ = 7 $\times$ 109 M⊙), late-type (Scd)
spiral galaxy in the outskirts of the M81 group at a distance of 3.2 Mpc
(Carlin et al., 2016). It may be infalling for the first time into the M81
group (Williams et al., 2013) with M81 itself lying $\sim$1 Mpc away. It has a
satellite galaxy DDO 44 (MV = –12.5), and recently a smaller dwarf, called
MADCASH-1 (MV = –7.7), was discovered by (Carlin et al., 2016). Although DDO
44 shows signs of interacting Carlin et al. (2019), NGC 2403 itself reveals a
remarkably undisturbed HI disk (de Blok et al., 2008; Williams et al., 2013).
This, and its relative isolation, suggests it is a galaxy that has not
undergone a strong tidal interaction nor experienced ram pressure stripping.
Although bulgeless, it does host a nuclear star cluster (Lira et al., 2007)
and it reveals an extended stellar structure at large radii which is either an
extension of the thick disk or the presence of a faint stellar halo (Barker et
al., 2012). It is also an important Cepheid variable calibrator galaxy (Saha
et al., 2006).

NGC 2403 has some global properties similar to those of the bulgeless Local
Group galaxy M33 (Scd, MV = –19.4). M33 contains a large population of
intermediate-age star clusters, particularly at large galactocentric radii
(Chandar et al., 2002; Davidge, 2003; San Roman et al., 2010; Fan & de Grijs,
2014) with 85% of them kinematically-associated with the inner halo and 15%
associated with the disk (Chandar et al., 2002). Chandar et al. (2002) found
that the inner halo GCs in M33 reveal a significant age spread when compared
to those of the Milky Way. A key difference is that M33 has signatures of an
inner stellar halo but not an outer one (Gilbert et al., 2021). This may
indicate a relatively small contribution from accreted dwarf galaxies in M33’s
assembly history. In this context, we note at least one halo GC (HM33-B) which
Larsen et al. (2021) argued was accreted on the basis of its anomalous
chemical signature.

The GC system of NGC 2403 has been poorly studied to date with very little
work on it since the photographic plate study of Battistini et al. (1984). In
this work they identified 8 candidates as having colours typical of GCs, 6
star clusters with bluer colours and 5 with redder colours than typical Milky
Way GCs. Davidge (2007) also identified another half dozen GC candidates in
their infrared imaging and estimated their sizes. NGC 2403 does not appear in
the catalogue of Harris et al. (2013) listing galaxies with known GC systems
(it lists M33 to have a total of 50 GCs) and its total number of GCs is
unknown. Based on its total halo mass of 2–3$\times$1011 M⊙ (Li et al., 2020),
and the relation of Burkert & Forbes (2020), we predict a total GC system of
40–50 members. Based on a typical GC specific frequency for spiral galaxies,
over a range of luminosity, of SN $\sim$ 1 (see Lomelí-Núñez et al., 2021),
around 60 GCs are expected. As well as having an undefined number of GCs, we
are unaware of any published high quality spectra for confirmed GCs in this
nearby galaxy. Thus the fraction of disk vs halo GCs and whether its halo GCs
resemble those of old halo GCs of the Milky Way, or its younger accreted GCs
are unknown. The age-metallicity relation (AMR) of NGC 2403’s GC system, which
potentially holds important clues on the galaxy’s assembly history, is
currently undefined.

Here we obtain Keck Cosmic Web Imager (KCWI) spectra for 9 GC candidates
around NGC 240 finding 7 to be confirmed GCs. Along with one other GC in NGC
2403 observed using HIRES on the Keck telescope, bringing the total to 8, we
discuss their stellar population properties and compare them to those of
SMC/LMC, M33 and the Milky Way. At a distance of 3.2 Mpc (Davidge, 2003), 1"
corresponds to 15 pc.

Figure 1: Difference between the $i$ band magnitudes measured on a 30s HSC
exposure of NGC 2403 with 3-pixel and 12-pixel apertures, $i_{3}-i_{12}$, as a
function of the i band PSF magnitude. Point sources occupy the locus at
$i_{3}-i_{12}\sim 0.35$. More extended objects containing excess flux
(relative to the PSF) exhibit larger $i_{3}-i_{12}$ values, with those between
0.5 and 0.9 (dashed lines) the best candidates for GCs. We highlight as blue
dots the two dozen candidate GCs selected based on all of our criteria, open
green squares highlight the six candidate GCs with matches to our HSC catalog,
and the orange diamond highlights the new candidate, JC15, that we followed up
spectroscopically (see section 2.2). Figure 2: Gaia parameters “astrometric
excess noise” (upper panel) and “BP/RP excess” (lower panel) as a function of
Gaia G magnitudes for objects in our HSC imaging. The curves in each panel are
the GC candidate selection criteria derived by Hughes et al. (2021) to select
Cen A clusters. Symbols are as in Figure 1.  Table 1: NGC 2403 Globular
Cluster Candidate Properties

ID | $g$ | $g-i$ | Source | Rp | Size
---|---|---|---|---|---
| (mag) | (mag) |  | (arcmin) | (pc)
C1 | 19.1 | 0.15 | PS | 5.3 | 3.5
C4 | 18.2 | 0.91 | DR15 | 5.3 | 2.0
D2 | 18.0 | 0.68 | DR15 | 8.6 | –
D6 | 19.3 | 0.95 | DR7 | 14.3 | 3.6
F1 | 19.1 | 0.88 | DR7 | 12.4 | 5.0
F16 | 19.5 | 1.01 | DR15 | 7.9 | 2.2
JD1 | 19.0 | 0.81 | DR7 | 6.7 | –
JD2 | 19.8 | 0.86 | DR15 | 80 | –
JC15 | 19.8 | 1.04 | HSC | 4.2 | –
F46 | 18.3 | 0.85 | DR15 | 5.0 | 4.4

Notes: Source of photometry, i.e. PanSTARRS, SDSS or Hyper Suprime-Cam
imaging. Rp is the projected radius from the centre of NGC 2403 (JD2 lies
close to DDO 44), with 1 arcmin equivalent to 900 pc. Sizes for D6, F1 are
from Davidge (2007), C1, C4, F16 from this work and F46 from L21.

##  2 Globular Cluster Candidates

Our GC candidates for follow-up spectroscopy have been taken from the
literature and supplemented by our own data.

###  2.1 Literature Candidates

From Battistini et al. (1984) we include their objects C1, C4, F1 and F16 and
from Davidge (2007) we include D2 and D6. According to Davidge (2007), several
GC candidates are partially resolved in their near-IR imaging. They measured
extended sizes for D2, D6, F1, F16 and F46. The candidates C1, C4, F16 and F46
are clearly resolved GCs in archival Hubble Space Telescope Advanced Camera
for Surveys (HST/ACS) imaging. We have estimated the effective radius of each
of these GCs after subtracting in quadrature the ACS point spread function
(based on several; stars in the image). Our radii are listed in Table 1, with
F46 from Larsen et al. (2021). While we find a similar size for F16 (2.2 vs
2.7 pc), we can rule out F46 having a size of 10.4 pc, as found by Davidge
(2007), based on the ACS imaging. For D2 they estimated a small but extended
size of 1.8 pc. We also examined the other candidates listed in Battistini et
al. (1984) with available HST/ACS imaging but concluded that are unlikely to
be GCs.

Figure 3:  A 28’ $\times$ 28’ colour SDSS image of NGC 2403 taken from
wikisky. The globular cluster candidates studied in this work are indicated
via open circles. They are labelled appropriately and colour coded by their
recession velocity (RV). We note that NGC 2403 has an RV of 133 km/s and the
receding side of the galaxy stellar/HI disk (i.e. higher RVs) is the SE side.
A 14.7’ $\times$ 14.7’ inlay is included to indicate the position of JD2 which
is $\sim 1.38$ deg North-North-West of NGC 2403 near the dwarf galaxy DDO 44.
At 3.2 Mpc, 1’ = 0.93 kpc. We find D2 and JD2 to be foreground stars.  Table
2: NGC 2403 Globular Cluster Candidate Spectroscopic Observations

ID | RA | Dec | RV | S/N | Notes
---|---|---|---|---|---
| (2000) | (J2000) | (km/s) |  |
C1 | 07:36:10.40 | +65:39:35.0 | 72 | 58 | Disk
C4 | 07:36:04.43 | +65:33:59.8 | 158 | 62 | Disk
D2 | 07:37:26.99 | +65:28:20.9 | 16 | 64 | MW Star
D6 | 07:35:05.74 | +65:45:27.4 | 165 | 22 | Halo
F1 | 07:35:12.43 | +65:43:15.8 | 210 | 45 | Halo
F16 | 07:38:01.52 | +65:33:00.6 | 165 | 43 | Disk
JD1 | 07:37:52.89 | +65:34:07.6 | 145 | 46 | Halo
JD2 | 07:33:34.58 | +66:56:29.8 | 0 | 12 | MW Star
JC15 | 07:36:18.55 | +65:38:33.0 | 96 | 8 | Disk
F46 | 07:36:29.16 | +65:40:33.5 | 140 | – | Halo

Notes: Radial velocity and S/N per Å are measured from our KCWI spectra. The
radial velocity of F46 was measured from a HIRES spectrum by L21. Typical
velocities uncertainties are on the order of 10 km s-1.

###  2.2 New Candidates

We used the characteristics of the candidates from Battistini et al. (1984)
and Davidge (2007) to help select additional GC candidates for spectroscopic
follow-up. In particular, using SDSS DR7 we searched for slightly resolved
objects with $r$-band Petrosian radius $<2$ arcsec, $i$-band magnitude in the
range 17 $<$ $i$ $<$ 19, colour in the range 0.7 $<$ $g-i$ $<$ 1.1 and 2.0 $<$
$u-z$ $<$ 3.5. Two of these candidates were targeted for follow-up
spectroscopy, which are named JD1 and JD2.

In the case of JD2, it was not selected to be a GC candidate of NGC 2403 as it
lies at a projected distance of $\sim$75 kpc (and hence greater in physical
distance), but it does lie near the dwarf satellite galaxy DDO 44 and might be
associated with it (given its mass, a couple of GCs might be expected).

We selected further candidates from imaging data obtained with Hyper Suprime-
Cam (HSC) on the Subaru 8.2m telescope. These data were gathered as part of
the Magellanic Analogs Dwarf Companions and Stellar Halos (MADCASH) survey, a
deep, wide-area search for dwarf satellites of nearby Magellanic Cloud
analogs. In addition to the deep exposures of NGC 2403 first presented in
Carlin et al. (2016), short (30s) exposures on each field were also taken to
prevent bright GC candidates from saturating. Here we use these g and i band
images taken on 2016/2/9, with seeing of $0.89\arcsec$ and $0.59\arcsec$,
respectively. After combining the g and i band source catalogs, we matched
them to Gaia EDR3 (Gaia Collaboration et al., 2016, 2021), selecting the best
match within $1\arcsec$ of each source. We then selected all objects with
magnitudes $17<i<20.5$, colours $0.5<(g-i)<1.3$, and at least $4\arcmin$ from
the centre of NGC 2403 (to avoid the most crowded regions of the images). We
further removed all objects whose Gaia proper motions or parallaxes are
measured with $>3\sigma$ significance, as GCs at $\sim 3$ Mpc distances should
not have measurable proper motions or parallaxes.

After these initial selection criteria, we explored aperture magnitudes from
the HSC data, and quality parameters from Gaia that are known to efficiently
select extended objects. Because we have a sample of known NGC 2403 GCs, we
can confirm that any selection criteria we implement effectively select known
GCs. Figure 1 shows the difference between the HSC i band magnitudes measured
in apertures of 3-pixel and 12-pixel radii, which we label $i_{3}-i_{12}$, as
a function of PSF magnitude. Objects that are very extended (i.e., galaxies)
will have a lot of flux beyond their 3-pixel radius, and thus have high values
of $i_{3}-i_{12}$, while stars should have a small amount of flux beyond a
3-pixel radius, which will be primarily determined by the extendedness of the
PSF (and thus roughly the same for all stars). The locus of stars is at
$i_{3}-i_{12}\sim 0.35$ in Figure 1. Extended galaxies will be well above this
locus, while GCs, which are only slightly extended, will have intermediate
flux ratios in this figure.

We note that 3 candidates do not appear in Figure 1. They are: JD1 which is
undetected because it lies in a crowded region of the NGC 2403 disk, JD2 which
is beyond the HSC field of view and F1 as it does not have reliable
measurements from Gaia (it nonetheless appears extended according to its
measured $i_{3}-i_{12}$ value). Candidate D2 has flux ratios in this figure
consistent with being a point source, and also a large measured proper motion
and is thus likely a foreground star.

Figure 2 displays the Gaia parameters “astrometric excess noise” (AEN) and
“BP/RP excess”, which have been shown by Hughes et al. (2021) to effectively
select objects around Cen A that are slightly more extended than the PSF
(i.e., candidate GCs), as a function of Gaia G magnitude. We overlay and adopt
the curves from Equations 1 and 2 from Hughes et al. to separate candidate
extended objects from point-like sources.

While this selection yields two dozen candidates, upon visual inspection, we
found many of these are obvious background galaxies. Additional work is
therefore required to create a comprehensive list, over the entire luminosity
function and spatially, of the 50 plus GC candidates expected around NGC 2403.

For the best candidate, which we name JC15, we have obtained a spectrum (see
below). JC15 lies close to the confirmed GCs D6 and F16 in the selection
figures. Most importantly, JC15 shows signs of extendedness in $i_{3}-i_{12}$,
but does not have extremely large flux ratios. It appears round in the i band
HSC images, with some hints of semi-resolved faint stars at the outskirts.

###  2.3 Candidates for Follow-up Spectroscopy

Our final sample of GC candidates for which we obtain spectra are given in
Table 1. It lists the g band magnitude and $g--i$ colour taken from SDSS or
HSC imaging. The quoted colours are not corrected for Galactic extinction nor
internal extinction within NGC 2403. Based on their projected positions on the
galaxy disk, JC15 may experience the most internal extinction (it is the
reddest object) and D6 the least. We also include in Table 1 (and subsequent
tables) the confirmed GC F46. This Battistini et al. (1984) GC candidate, was
subsequently confirmed by Larsen et al. (2021) (hereafter L21) using the HIRES
instrument at the Keck Observatory.

Fig. 3 shows an image of NGC 2403 with the locations of our GC candidates
indicated. Other than JD2, all of our GC candidates follow a flattened disk
structure along the major axis. Given the lack of a bulge in NGC 2403, we
expect confirmed GCs to be associated with either the disk or the inner halo.
In Fig. 7 we show a montage of SDSS, or if available HST, ‘postage stamp’
images of each GC candidate (including F46 by L21).

##  3 Data

Globular cluster candidates were observed using the KCWI imager slicer
(Morrissey et al., 2018) on the Keck II telescope over several different
nights under programs W140, U216, W024 and N195. In particular, they are D2
(2019/10/29), C4, D6, JD2 (2020/2/16), JC15 (2021/10/4), and F1, F16, JD1, C1
(2022/1/28).

Exposure times ranged from 10 to 40 mins under 1–2" seeing conditions. We used
the medium field-of-view and the BH3 grating (with a spectral resolution R =
9000) for C4, D2, D6, JD2. For C1, F1, F16 and JD1 we used the large field-of-
view and the BL grating (R = 900). Standard stars were observed in the same
setups as the target objects. Some lower S/N spectra were also obtained for
some of these candidates but they are not used in this current work.

Figure 4: KCWI spectra for each confirmed globular cluster as labelled.
Original spectra, shifted to rest wavelength, are shown in black and our model
fits in magenta. The key absorption lines used for the analysis are shown in
vertical lines (e.g., H$\beta$, Fe5015, Mgb and Fe5270), as labelled in the
top panel. Note that different KCWI grating setups were used in the
observations, and thus the grey area corresponds to wavelengths not covered by
the shorter grating, the BH3

.

The data were processed using the standard KCWI data reduction pipeline. We
took the non-sky subtracted, standard star calibrated cubes and cropped them
to the wavelength range common to all slices to use them for the remainder of
our analysis. The globular clusters were sky subtracted by taking an
appropriately sized region centred on each cluster and subtracting off an on-
chip offset region as ‘sky’. Where multiple exposures were taken on the same
object these were mean stacked to produce a single spectrum for each GC. Final
integration times for each object were: 1440s for C1, 2400s for C4, 1200s for
D02, 1200s for D06, 1200s for F1, 1440s for F16, 1200s for JD1, 900s for JD2
and 900s for JC15.

Table 2 lists the candidate coordinates, signal-to-noise ratio of each
spectrum and our measured radial velocity (with a barycentric correction
applied) with typical uncertainties of $\sim$10 km s-1. All of the measured
radial velocities are consistent with those expected from NGC 2403, however
they are also within the velocity range of Milky Way stars.

The GC candidates shown in Fig. 3 are colour coded according to their measured
radial velocity (RV). The general rotation of the stellar and HI disk from
Fraternali et al. (2002) shows that the SE side of NGC 2403 is receding at 250
km -1 while the NW side is ‘approaching’ us at 10 km s-1 (the maximum disk
rotation velocity is around 130 km s-1). The mean RV of NGC 2403 is 133 km
s-1. On the basis of measured velocities relative to the galaxy’s velocity
field we suggest that D6, F1 and JD1 (and F46) are GCs likely associated with
the halo rather than the disk of NGC 2403. Given these GCs all lie within a
projected distance of 13 kpc they may be more akin to inner halo GCs as seen
in the Milky Way’s GC system. The objects consistent with the general disk
rotation are C1, C4, F16 and JC15.

Although the dwarf galaxy DDO 44 reveals a tidal stream due to likely
interaction with NGC 2403 (Carlin et al., 2019) it has a recession velocity of
255 km s-1 and so JD2 with a velocity of $\sim$0 km s-1 is very unlikely to be
associated with DDO 44. We conclude that JD2 is a likely foreground star.
Candidate D2 has a blue colour, a near zero radial velocity, a similar flux in
3 and 12 pixels in our HSC imaging and Gaia parameters consistent with a Milky
Way star. So despite the extended size for D2 given by Davidge (2007), we
conclude that it is actually a foreground star.

The status of each GC candidate is summarised in Table 2, with 7 being
confirmed GCs and 2 as foreground stars. We will therefore no longer consider
these stars further. Fig. 4 shows the KCWI spectra for the final sample of 7
confirmed GCs (and highlights the different wavelength scales used for the
observations). The key absorption lines are indicated.

##  4 Stellar Population Analysis

Table 3: NGC 2403 Globular Cluster Stellar Populations

ID | Age | [Z/H] | [Fe/H] | M∗
---|---|---|---|---
| (Gyr) | (dex) | (dex) | (105 M⊙)
C1 | 0.4$\pm$0.2 | –0.61$\pm$0.29 | –0.4$\pm$0.50 | 0.5
C4 | 12.5$\pm$1.6 | –2.07$\pm$0.18 | –2.20$\pm$0.15 | 12
D6 | 7.8$\pm$1.5 | –1.25$\pm$0.04 | –1.20$\pm$0.15 | 2.8
F1 | 9.0$\pm$2.9 | –1.25$\pm$0.20 | –1.35$\pm$0.30 | 5.0
F16 | 9.3$\pm$2.2 | –0.71$\pm$0.29 | –1.15$\pm$0.20 | 4.1
JD1 | 8.6$\pm$2.9 | –1.24$\pm$0.31 | –1.60$\pm$0.30 | 4.5
JC15 | 10.8$\pm$2.1 | –1.41$\pm$0.28 | –1.50$\pm$0.20 | 2.9
F46 |  $\leq$13 | – | –1.70$\pm$0.03 | 11

Notes: See text for details of possible systematic effects on stellar
population parameters. F46 measured from a HIRES spectrum by L21.

To analyse the stellar populations of the 7 GCs, we follow the method outlined
in Ferré-Mateu et al. (2021) for compact stellar systems. Briefly, we use the
MILES Single Stellar Population (SSP) library (Vazdekis et al., 2010) models
with the BaSTI isochrones, considering templates that range from metallicities
of [Z/H] = –2.42 to +0.40 dex and that cover ages from 0.03 to 14 Gyr,
assuming a universal Kroupa IMF. The SSP models include different alpha
element ratios, from solar to super-solar (Vazdekis et al., 2016). The work of
Mendel et al. (2007) compared the SSP models of Vazdekis et al. to other SSP
models and found good consistency for Milky Way GCs. We run pPXF using
multiplicative polynomials and applying a regularization value that ensures
that the resulting star formation history is the smoothest possible while
maintaining a realistic fit. The best fit SSP model using pPXF (Cappellari,
2012) for each GC is overplotted in Fig. 4. For the results, we adopt the
means of the mass-weighted stellar population parameters. However, for the
[Fe/H] metallicity that is measured directly from the absorption line indices.
For the [$\alpha$/Fe] ratios we employ the Mgb and Fe5015 lines (which are
common to all spectra; these same pairs were employed by the SAURON survey of
early-type galaxies; Peletier et al. (2007)). We find that only JD1 has a
clear super-solar alpha element enhancement and, in this case we employ the
corresponding alpha-enhanced, rather than base, MILES models. Otherwise the
GCs are consistent with a solar alpha element ratio.

Table 3 presents the mean mass-weighted stellar populations that we measure
for the ages and metallicities with uncertainties. Quoted uncertainties are
determined using a Monte Carlo technique in which we have varied different
ingredients in the pPXF analysis, (e.g. polynomials, degree, regularization,
alpha vs. base models) to determine the overall uncertainties. Stellar masses
are calculated from $g$-band magnitudes and the resulting mass-to-light ratio
from the SSP analysis. We note the very young age of the cluster C1 (400 Myr)
and the strong Balmer absorption lines in its spectrum. Our SSP analysis
reveals no evidence for a hidden old stellar population for this GC. Although
we continue to refer to C1 as a GC, some prefer the ‘young massive cluster’
nomenclature.

For F46, we list the stellar population parameters determined independently by
Larsen et al. (2021), i.e. their assumed age and measured [Fe/H] metallicity
(they found [$\alpha$/Fe] = +0.15). Their assumed age, of 13 Gyr, can be
considered an upper limit. For the stellar mass we assume the same ratio as
JC15 (i.e. M/Lg $\sim$ 2) which gives a stellar mass of 11 $\times$ 105 M⊙ in
reasonable agreement with the dynamical mass calculated by L21 of 15 $\times$
105 M⊙.

Using the stellar parameters determined from our SSP analysis, we predict
$g-i$ colours for our GCs. These predicted colours are compared to the
measured colours, corrected for Galactic extinction (AV = 0.11), in Fig. 5.
The predicted and Galactic extinction-corrected colours are in reasonable
agreement, thus providing further confidence in our stellar population
analysis. Correction for internal dust extinction within NGC 2403 for the disk
GCs could improve the agreement further.

Before discussing our stellar population results it is worth reviewing the
potential biases and caveats in the object selection and data analysis. Both
the literature and our own selection of GC candidates are naturally biased
away from the inner disk due to the possible confusion with young star
clusters and HII regions. This might tend to bias our sample against the
selection of metal-rich disk GCs.

The stellar masses calculated for our sample of GCs (see Table 3) are all but
one above 105M⊙. With a mean GC mass of around 2$\times$105M⊙ for the Milky
Way’s GC system, this indicates that our sample is drawn from the most massive
or brightest (for comparison the GC turnover magnitude is Mg $\sim$ –7.2) of
the GCs in NGC 2403. We note that selecting the more massive GCs from the
Milky Way’s system would create a bias towards selecting accreted GCs (Mackey
& van den Bergh, 2005).

There is the possibility of contamination in our GC spectra by emission lines
associated with star formation in the disk of NGC 2403. If present, H$\beta$
emission would tend to fill in the H$\beta$ absorption line resulting in an
older inferred age if not corrected for. If contamination from emission lines
was present, we would expect to see the [OIII] line at 5007 Å – a visual
inspection reveals no clear evidence for [OIII]. Furthermore, there is no
evidence for emission lines from our pPXF analysis of the entire spectrum.
Thus we conclude that our spectra do not suffer from such contamination.

Moreover, the observed GCs may contain blue horizontal branch (BHB) stars.
These hot stars can give the impression of a young spectral age while having
limited effect on the derived metallcitity and alpha elements (Conroy et al.,
2018). The study of Mendel et al. (2007) examined the effects of BHB stars on
the derived stellar population parameters from integrated spectra and found a
limited impact when a large spectral range was used. We thus consider that the
ages of those GCs with a shorter baseline should be considered lower limits,
as without higher quality and/or longer baseline spectra we are unable to rule
out the presence of BHB stars.

Figure 5: Comparison of $g-i$ colours predicted from our single stellar
population (SSP) analysis vs the observed colours (corrected for Galactic
extinction) for NGC 2403 globular clusters. Magenta filled circles show disk
GCs and filled triangles show inner halo GCs. The plot focuses on the old, red
GCs: the young blue cluster C1 (with SSP $g-i$ of 0.2 and observed $g-i$ of
0.15) is not shown. The dashed line shows a one-to-one line. Corrections for
internal extinction would move data to bluer observed colours.

##  5 Results and Discussion

In this work we obtain spectra, radial velocities, stellar population
parameters (age, metallicity and alpha abundances) and stellar masses for 7
confirmed GCs associated with NGC 2403. One additional GC, F46, was previously
observed using HIRES at the Keck Observatory by Larsen et al. (2021) and we
include it in our subsequent analysis. They derived a radial velocity
(confirming association with NGC 2403), a low metallicity and a mildly
enhanced alpha element ratio. Previously, Battistini et al. (1984) managed to
obtain low quality spectra of 3 GC candidates (C3, C4 and F21). For C3 they
noted similarities to a A-type spectrum indicating an age of around $\leq$1
Gyr. For the only object in common with this study, C4, they described the
spectrum as being that of a classical GC (which we confirm). For F21 they
concluded it was a background galaxy.

Figure 6: Age-metallicity relation for NGC 2403 globular clusters. Magenta
filled circles show NGC 2403 disk GCs and filled triangles show inner halo
GCs. For GC F46 an age of 13 Gyr was assumed in L21. The green squares with
error bars show an intermediate-age GC and the mean value for several outer
halo GCs in M33. Red and blue squares (without error bars) show a selection of
LMC and SMC GCs respectively, with reliable ages and metallicites. Small black
circles show Milky Way GCs. The magenta line shows a representative chemical
enrichment model for the NGC 2403 GCs. The NGC 2403 GCs are more metal-poor on
average at a given age than those in the Milky Way but similar to those in the
SMC.

The stellar populations, and possible biases/caveats, for the Keck-observed
GCs are summarised in Section 4 and Table 3. Bearing in mind that 8 GCs out of
a possible GC system of around 50 may not be representative, we discuss our
sample of high-mass disk/halo GCs in the context of GC systems in other nearby
galaxies below.

Our data indicate that the GC system of NGC 2403 was formed over an extended
period of time from early in the Universe until today. The GCs have near solar
alpha-element ratios with one presenting super-solar enhancement. Compared to
the two subpopulations of the Milky Way’s GC system, the old GCs studied here
are all relatively metal-poor ([Fe/H] $\leq$ –1); only the very young cluster
C1 is metal-rich. We note that Davidge (2003) also measured a low metallicity
of [Fe/H] = –2.2 for field stars in the outer halo (projected $\sim$22 kpc
along the minor axis, beyond our most distant GC) of NGC 2403. This
metallicity is comparable to those for the oldest GCs in our sample.

In Fig. 6 we show the ages and [Fe/H] metallicities for our NGC 2403 GCs,
along with those of the Milky Way, the Magellanic Clouds and M33 (a possible
analogue galaxy to NGC 2403). For M33 we show a confirmed intermediate-aged
(2–5 Gyr) GC from (Chandar et al., 2006) which has a metallicity of [Fe/H] =
–0.65$\pm$0.16 (Sarajedini et al., 2000). We also show the mean values for a
dozen old M33 GCs (i.e. age = 10.35$\pm$0.71 Gyr and [Fe/H] = –1.12$\pm$0.09)
from Beasley et al. (2015). We refer the reader to Larsen et al. (2021) for
arguments that some of the Beasley et al. objects may actually be foreground
stars (their exclusion does not significantly change the mean values). For the
Magellanic Clouds we take the reliable ages and metallicities from the
appendix of Horta et al. (2021), i.e. those with a confidence code of 1. We
take ages and metallicities for Milky Way GCs from Forbes (2020). In the Milky
Way, the in-situ formed bulge and disk GCs formed over a short time period
(1-2 Gyr) at early times ($\sim$ 13 Gyr ago), whereas the younger GCs are all
the result of ex-situ formation and subsequent accretion of their host dwarf
galaxy (Forbes, 2020).

Our observed NGC 2403 GCs in Fig. 6 are coded by whether we have assigned them
to the halo or to the disk of NGC 2403 on the basis of their relative
velocities (see Table 2). We find no clear distinction between disk and halo
GCs in terms of their average age or metallicity. The (disk) GC F46 is shown
with an upper age limit of 13 Gyr as assigned by L21. Given that NGC 2043 has
little or no bulge (similar to M33), we do not expect bulge GCs to be present,
as they are in the Milky Way. At a given age, the NGC 2403 GCs observed in
this study have metallicities most similar to those of the SMC, slightly less
than those of M33 and the LMC, and systematically lower than GCs accreted from
dwarf galaxies onto the Milky Way. In terms of the age distribution, the
Magellanic Clouds, M33 and NGC 2403 all appear to have formed star clusters
from early times ($\sim$ 13 Gyr ago) until just a few Gyr ago. We remind the
reader of C3 in NGC 2403, which Battistini et al. (1984) claimed it has an
A-type spectrum indicative of a $\leq$1 Gyr old GC. Furthermore massive ($\sim
10^{5}$ M⊙), young ($\sim$0.1 Gyr) star clusters have been identified by
Larsen & Richtler (1999) to be forming up to the present day in NGC 2403.

To further discuss the age-metallicity distribution of our observed GCs, we
also show in Fig. 6 the age-metallicity relation (AMR) for a leaky-box
chemical enrichment model. The chemical enrichment models assume non-enriched
gas 13.5 Gyr ago and an effective yield of $p$ (i.e. elements returned to the
ISM for subsequent star formation after accounting for outflows of enriched
material), which is related to stellar metallicity by the equation:

[Fe/H] = - $p$ ln ($t$/13.5)

Such a model provides a good representation for the GCs from disrupted
satellites in the Milky Way. Indeed, this aspect of dwarf galaxy GC systems
was used by Kruijssen et al. (2019) and Forbes (2020) to associate individual
Milky Way GCs with disrupted satellites. The yields for the five largest
disrupted satellites, including the Sgr dwarf, were found to range from 0.22
to 0.35. The model shown in Fig. 6 has $p$ = 0.1 and is very similar to the
one derived from the stars in the SMC (M∗ = 0.4 $\times$ 109 M⊙) from Leaman
et al. (2013).

The steepness of an AMR (i.e. the metallicity at a given age), as represented
by the yield, is primarily driven by the mass of the host galaxy. Given that
the AMR for NGC 2403 appears to be lower than that of dwarf galaxies accreted
onto the Milky Way and perhaps also the GCs of M33 (which has a similar
luminosity to NGC 2403), other factors could be at play. One factor that may
act to lower an AMR is the outlfow of enriched material. In a study of nearby
spirals and dwarf galaxies, Dalcanton (2007) concluded that "Metal-enriched
outflow is therefore the only viable mechanism for producing galaxies with low
effective yields." The presence of an extended ‘HI halo’ around NGC 2403 led
Fraternali et al. (2002) to suggest that it may be the signature of a galactic
fountain. This outflow from the disk to the halo may contribute to the low
effective yield as argued by Dalcanton (2007). Furthermore, there is a step
change in the effective yield to lower values at rotation speeds of
V${}_{rot}\leq 150$ km s-1 (Garnett, 2002). NGC 2043, with Vrot $\sim$ 130 km
s-1 falls close to this key transition.

Cosmological simulations predict that galaxies of NGC 2403’s mass have not
undergone significant accretion (e.g. in the model of Choksi & Gnedin (2019))
and only in-situ formed GCs are expected. Indeed there are several arguments
against NGC 2403 having experienced a massive merger event in the past, which
would mean few, if any, accreted GCs. NGC 2403 is relatively isolated and may
only be infalling into the low mass M81 group for the first time (there is no
evidence of ram pressure stripping). It also reveals a remarkably uniform
regularly rotating HI disk (de Blok et al., 2008). Its old stellar disk has a
uniform metallicity of [Fe/H] $\sim$ –1 (consistent with our most metal-rich
GCs), indicating that it is well-mixed, and has an unbroken surface brightness
profile (Williams et al., 2013). This suggests a relatively undisturbed galaxy
despite its probable interaction with the dwarf satellite DDO 44 (Carlin et
al., 2021). Davidge (2003) found evidence for AGB stars (ages $\leq$ 1 Gyr)
beyond the disk of NGC 2403 and argued that they formed in-situ and not from
accretion. We also note that F46 does not have the anomalous chemical
signature indicative of an accreted GC (Larsen et al., 2021). While we cannot
rule out accreted GCs, our sample of GCs is likely formed, over many Gyr, from
in-situ gas.

##  6 Conclusions

In this work we studied a sample of relatively high mass globular clusters
(GCs) associated with the low mass, bulgeless galaxy NGC 2403. Based on GC
candidates in the literature, and new candidates selected from SDSS and our
Hyper Suprime-Cam imaging, we obtained spectra of nine candidates using the
KCWI instrument at the Keck Observatory. We confirm seven of them to be GCs
associated with NGC 2403, with the remaining two likely to be foreground
stars. Our results are supplemented with one additional GC observed using
HIRES, also from Keck, by Larsen et al. (2021).

By comparing the radial velocities of the eight confirmed GCs with the overall
rotation of the HI disk, we assign four GCs to the disk and four to the halo.
Single stellar population model fits indicate that the GCs were formed over
many Gyr (with the caveat that the presence blue horizontal branch stars can
make GCs appear much younger than they are). We find most to be consistent
with solar alpha element ratio. However, the old GCs are all found to be
metal-poor, i.e. [Fe/H] $\leq$ –1. The one young, massive cluster we obtain a
spectrum for is more metal-rich. Cosmological simulations predict that few, if
any, accreted GCs will be found in galaxies of the mass of NGC 2403. Based on
the lack of disturbance in the stellar and HI disk, and its relative
isolation, both the disk and halo GCs of NGC 2403 likely formed over an
extended period from in-situ gas.

Our NGC 2403 GCs are systematically lower in metallicity at a given age
compared to the Milky Way’s GC system and perhaps also to the GCs in M33.
However, they exhibit a similar age-metallicity relation to the GCs of the
Small Magellanic Cloud. The age-metallicity relation of the NGC 2403 GCs can
be approximated by a leaky-box chemical enrichment model with a similar
effective yield than that inferred for the SMC stars. Outflows of enriched
material may explain the relatively low yield inferred for NGC 2403’s GC
system.

## Acknowledgements

We thank. S. Larsen for supplying information regarding the globular cluster
F46 prior to publication and for his comments on our paper. We thank the
MADCASH team for the original HSC observations and D. Sand for his comments.
We thank the referee for several suggestions that have improved this paper.
AFM has received support from the Severo Ochoa Excellence scheme
(CEX2019-000920-S) and through the Postdoctoral Junior Leader Fellowship
Programme from ’La Caixa’ Banking Foundation (LCF/BQ/LI18/11630007). AJR was
supported by National Science Foundation grant AST-1616710 and as a Research
Corporation for Science Advancement Cottrell Scholar. JLC acknowledges support
from National Science Foundation grant AST-1816196.

This work was supported by a NASA Keck PI Data Award, administered by the NASA
Exoplanet Science Institute. Data presented herein were obtained at the W. M.
Keck Observatory from telescope time allocated to the National Aeronautics and
Space Administration through the agency’s scientific partnership with the
California Institute of Technology and the University of California. The
Observatory was made possible by the generous financial support of the W. M.
Keck Foundation. The authors wish to recognize and acknowledge the very
significant cultural role and reverence that the summit of Maunakea has always
had within the indigenous Hawaiian community. We are most fortunate to have
the opportunity to conduct observations from this mountain.

## Data Availability

Keck data are available from the Keck Observatory Archive (KOA) at
https://www2.keck.hawaii.edu/koa/public/koa.php. Sloan Digital Sky Survey
(SDSS) photometry is available at http://skyserver.sdss.org/dr7/en/home.aspx.
HST data are available from the MAST archive at
https://archive.stsci.edu/missions-and-data/hst.

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##  Appendix A Globular cluster candidate imaging

Images, from HST or SDSS, of our globular cluster candidates studied in this
work, along with the confirmed GC (F46) of Larsen et al. (2021). We find two
candidates (D2 and JD2) to be foreground stars.

Figure 7: A montage of the globular cluster candidates studied in this work
taken from SDSS or, if available, from HST imaging. Globular cluster IDs are
labelled. Each image postage stamp covers a field of view of 25.6" $\times$
25.6". At 3.2 Mpc, 1" = 15 pc. We conclude that D2 and JD2 are foreground
stars (see text for details).