#  On Beurling Measure Algebras

Ross Stokke111 This research was partially supported by an NSERC grant.

###### Abstract

We show how the measure theory of regular compacted-Borel measures defined on
the $\delta$-ring of compacted-Borel subsets of a weighted locally compact
group $(G,\omega)$ provides a compatible framework for defining the
corresponding Beurling measure algebra ${\cal M}(G,\omega)$, thus filling a
gap in the literature.

Primary MSC code: 43A10 Secondary MSC codes: 22D15, 43A05, 43A20, 43A60, 28C10
Key words and phrases: Weighted locally compact group, group algebra, measure
algebra, Beurling algebra

Throughout this article, $G$ denotes a locally compact group and
$\omega:G\rightarrow(0,\infty)$ is a continuous weight function satisfying

$\omega(st)\leq\omega(s)\omega(t)\ \ (s,t\in G)\qquad{\rm
and}\qquad\omega(e_{G})=1;$

the pair $(G,\omega)$ is called a weighted locally compact group. Let
$\lambda$ denote a fixed Haar measure on $G$, with respect to which the group
algebra $L^{1}(G)$ and $L^{\infty}(G)=L^{1}(G)^{*}$ are defined in the usual
way. The Beurling group algebra, $L^{1}(G,\omega)$, is composed of all
functions $f$ such that $\omega f$ belongs to $L^{1}(G)$, with
$\|f\|_{1,\omega}:=\|\omega f\|_{1}$ and convolution product. If ${\cal S}(G)$
is a closed subspace of $L^{\infty}(G)$, $\psi\in{\cal S}(G,\omega^{-1})$
exactly when ${\psi\over\omega}\in{\cal S}(G)$; putting
$\|\psi\|_{\infty,\omega^{-1}}=\left\|{\psi\over\omega}\right\|_{\infty}$,
${\cal S}(G,\omega^{-1})$ is a Banach space and $S:{\cal
S}(G,\omega^{-1})\rightarrow{\cal S}(G):\psi\mapsto{\psi\over\omega}$ is an
isometric linear isomorphism. The Beurling group algebra $L^{1}(G,\omega)$ has
become a classical object of study that has received significant research
attention over the years: see the monographs [3, 11, 15] and the references
therein; a sample of relevant articles include [5, 7, 8, 9, 17, 18, 20]. When
$\omega$ is the trivial weight $\omega\equiv 1$ — the “non-weighted case” —
$L^{1}(G,\omega)=L^{1}(G)$, the study of which is intimately linked with the
measure algebra $M(G)$ of complex, regular, Borel measures on $G$, which
contains $L^{1}(G)$ as a closed ideal.

The above definition of $L^{1}(G,\omega)$ is valid for any weight $\omega$. As
in the non-weighted case, it is desirable to have a Beurling measure algebra
$M(G,\omega)$ that shares the same relationship with $L^{1}(G,\omega)$ that
$M(G)$ shares with $L^{1}(G)$. In the literature, $M(G,\omega)$ is usually
defined as the collection of all complex regular measures $\nu$ defined on
${\mathfrak{B}}(G)$, the $\sigma$-algebra of Borel subsets of $G$, such that
$\int\omega(t)\,d|\nu|(t)<\infty,$ and the identification
$M(G,\omega)=C_{0}(G,\omega^{-1})^{*}$ through
$\langle\nu,\psi\rangle_{\omega}=\int\psi\,d\nu$ is required. This implies
that the dual map, $S^{*}$, of the isometric isomorphism
$S:C_{0}(G,\omega^{-1})\rightarrow C_{0}(G)$ is itself a linear isometric
isomorphism of $M(G)$ onto $M(G,\omega)$. Validity of this definition of
$M(G,\omega)$ thus requires that for each $\mu\in M(G)$, $\nu=S^{*}\mu\in
M(G,\omega)$ is a complex Borel measure defined on all of ${\mathfrak{B}}(G)$
— the near-universal requirement of “Borel measures” in abstract harmonic
analysis — satisfying

$\displaystyle\int\psi\,d\nu=\langle\nu,\psi\rangle_{\omega}=\left\langle\mu,{\psi\over\omega}\right\rangle=\int{\psi\over\omega}\,d\mu\qquad(\psi\in
C_{0}(G,\omega^{-1})).$ (1)

However, when $\omega$ is not bounded away from zero, it can happen that no
such complex measure on ${\mathfrak{B}}(G)$ exists.

To see this, consider $(G,\omega)$ where $G=(\mathbb{Z},+)$ and
$\omega(n)=2^{-n}$ $(n\in\mathbb{Z})$, and assume the above definition of
$M(G,\omega)$ is sound. Since
$\mu_{1},\mu_{2}\in\ell^{1}(\mathbb{Z})^{+}=M(\mathbb{Z})^{+}$ and
$\mu=\mu_{1}-\mu_{2}\in M(\mathbb{Z})$, where

$\displaystyle{\mu_{1}(n)=\left\\{\begin{array}[]{ll}2^{-n}&\mbox{ $n\in
2\mathbb{N}$}\\\ 0&\mbox{ otherwise}\end{array}\right.}\qquad{\rm
and}\qquad\displaystyle{\mu_{2}(n)=\left\\{\begin{array}[]{ll}2^{-n}&\mbox{
$n\in\mathbb{N}\backslash 2\mathbb{N}$}\\\ 0&\mbox{
otherwise}\end{array}\right.},$

$\nu_{1}=S^{*}(\mu_{1})$, $\nu_{2}=S^{*}(\mu_{2})$, and
$\nu=S^{*}(\mu)=\nu_{1}-\nu_{2}$ are then required to be complex measures on
${\mathfrak{B}}(G)=\wp(\mathbb{Z})$ satisfying (1). Hence, for each
$n\in\mathbb{Z}$,

$\displaystyle{\nu_{1}(\\{n\\})=\int\chi_{\\{n\\}}\,d\nu_{1}=\left\langle\mu_{1},{\chi_{\\{n\\}}\over\omega}\right\rangle=\left\\{\begin{array}[]{ll}1&\mbox{
$n\in 2\mathbb{N}$}\\\ 0&\mbox{ otherwise}\end{array}\right.}\quad{\rm
and}\quad\displaystyle{\nu_{2}(\\{n\\})=\left\\{\begin{array}[]{ll}1&\mbox{
$n\in\mathbb{N}\backslash 2\mathbb{N}$}\\\ 0&\mbox{
otherwise}\end{array}\right.};$

therefore,
$\displaystyle\nu_{1}(2\mathbb{N})=\sum_{k\in\mathbb{N}}\nu_{1}(\\{2k\\})=+\infty$
and $\displaystyle\nu_{2}(\mathbb{N}\backslash
2\mathbb{N})=\sum_{k\in\mathbb{N}}\nu_{2}(\\{2k-1\\})=+\infty$. Thus,
$\nu_{1}$, $\nu_{2}$ do not map into $\mathbb{C}$. Moreover, (although
$\nu_{1}$, $\nu_{2}$ can be viewed as positive measures), if
$\nu=\nu_{1}-\nu_{2}$ were a measure, additivity would give

$\nu(\mathbb{N})=\nu(2\mathbb{N})+\nu(\mathbb{N}\backslash
2\mathbb{N})=\nu_{1}(2\mathbb{N})-\nu_{2}(\mathbb{N}\backslash
2\mathbb{N})=\infty-\infty.$

We conclude that functionals in $C_{0}(G,\omega^{-1})^{*}$ cannot necessarily
be identified with complex Borel measures in the standard sense. It is perhaps
for this reason that many authors assume the additional condition $\omega\geq
1$, since this guarantees containment of $M(G,\omega)$ in $M(G)$ and, thus,
the essential properties of $M(G)$ also hold for $M(G,\omega)$, e.g., see [3].
Letting $\mathfrak{S}(G)$ denote the $\delta$-ring of “compacted-Borel sets” —
i.e., the $\delta$-ring of all Borel subsets of $G$ with compact closure — a
compacted-Borel measure on $G$ is a countably additive complex-valued function
on $\mathfrak{S}(G)$ in the sense of [4, Definitions II.1.2 and II.8.2]222In
[4], for the sake of brevity, the authors refer to compacted-Borel measures
simply as Borel measures. To our knowledge, with the exception of [4], Borel
measures in abstract harmonic analysis are always defined on
$\mathfrak{B}(G)$.. For non-compact $G$, there are positive regular measures
$\mu,\nu$ on ${\mathfrak{B}}(G)$ such that $\mu(G)=\nu(G)=\infty$ (e.g., Haar
measures), and therefore $\mu-\nu$ is not defined on ${\mathfrak{B}}(G)$;
however, these same measures are real-valued on $\mathfrak{S}(G)$, so
$\mu-\nu$ is well-defined on $\mathfrak{S}(G)$. This is one benefit to
studying measure theory over $\mathfrak{S}(G)$, rather than on all of
${\mathfrak{B}}(G)$.

The purpose of this article is to show that the theory of complex regular
compacted-Borel measures, as developed in [4] (also see paragraph two of the
“Notes and Remarks” section of Chapter II of [4] for additional references),
can be used to provide a rigorous definition of $M(G,\omega)$, thus providing
a solid foundation for all the papers in which $M(G,\omega)$ is employed
without the requirement that $\omega\geq 1$; moreover, we hope this reduces
the number of instances in which the $\omega\geq 1$ assumption is required
going forward. To stress that we are using the theory of complex regular
compacted-Borel measures, we will use the notation ${\cal M}(G,\omega)$ —
inspired by [4] — rather than $M(G,\omega)$. Beyond identifying the correct
collection of measures to employ, work is required to establish the needed
theory. As measure theory can be quite finicky in general; because the study
of compacted-Borel measures introduces different technicalities than those
encountered in the Borel measure situation; and because a lot of research
already depends on the results found herein, we have included a careful
treatment of our development of ${\cal M}(G,\omega)$. There are numerous
detailed classical expositions of the basic theory $M(G)$, and we believe the
same is required for ${\cal M}(G,\omega)$.

We restrict ourselves to developing only the most standard properties of
${\cal M}(G,\omega)$: we provide a careful definition of its elements and show
that with convolution product it is a dual Banach algebra containing a copy of
the Beurling group algebra $L^{1}(G,\omega)$ as a closed ideal. Beyond this,
we only show that ${\cal M}(G,\omega)$ embeds via a strict-to-weak∗ continuous
isometric isomorphism as a subalgebra of the universal enveloping dual Banach
algebra of ${\cal L}^{1}(G,\omega)$, $WAP(L^{\infty}(G,\omega))^{*}$, a result
needed in [12]. The inspiration for this paper was our need to work with
${\cal M}(G,\omega)$ in [12].

##  1 ${\cal M}(G,\omega)$: definition and basic properties

Unless explicitly indicated otherwise, all references are to statements in §s
1,2,5,7-10 of Chapter II and §10 of Chapter III of [4]. We will mostly adhere
to the notation found therein. In particular, ${\cal M}(G)$ is the linear
space composed of all regular complex compacted-Borel measures on $G$ (§s II.8
and III.10) and ${\cal M}_{r}(G)$ is the Banach space of bounded measures in
${\cal M}(G)$ (§s II.1 and II.8). Let $\mathfrak{C}(G)$ denote the directed
set of compact subsets of $G$, and denote the space of continuous functions on
$G$ with compact support by $C_{00}(G)$, the space of continuous functions on
$G$ vanishing at infinity by $C_{0}(G)$, and the space of continuous functions
on $G$ supported on $K\in\mathfrak{C}(G)$ by $C_{K}(G)$; unless the context
requires otherwise, these spaces are taken with the uniform norm
$\|\cdot\|_{\infty}$.

######  Remark 1.1.

(a) Let $\mu\in{\cal M}(G)$. A Borel subset $A$ of $G$ belongs to ${\cal
E}_{\mu}$ if $A$ is contained in some open set $U$ such that

$\sup\\{|\mu|(A^{\prime}):A^{\prime}\in\mathfrak{S}(G)\ {\rm and}\
A^{\prime}\subseteq U\\}<\infty;$

${\cal E}_{\mu}$ is a $\delta$-ring containing $\mathfrak{S}(G)$ and, for
$A\in{\cal E}_{\mu}$, putting

$\displaystyle\mu_{e}(A):=\lim_{C}\mu(C)\quad{\rm where}\
C\in\mathfrak{C}(G),\ C\subseteq A,$ (2)

we obtain a complex measure on ${\cal E}_{\mu}$ extending $\mu$, called the
maximal regular extension of $\mu$ (II.8.15). Observe that any Borel subset of
a set in ${\cal E}_{\mu}$ is also in ${\cal E}_{\mu}$, from which it readily
follows that $h\chi_{E}$ is locally $\mu_{e}$-measurable whenever $E\in{\cal
E}_{\mu}$ and $h$ is a Borel-measurable function on $G$.

(b) When $\mu\in{\cal M}_{r}(G)$, ${\cal E}_{\mu}={\mathfrak{B}}(G)$ and
$\mu_{e}\in M(G)$, where $M(G)$ denotes the usual measure algebra of regular
complex Borel measures $\mu:{\mathfrak{B}}(G)\rightarrow\mathbb{C}$, e.g., see
[2, 6, 14]. Thus, the measures in ${\cal M}_{r}(G)$ are in one-to-one
correspondence with measures in $M(G)$ via $\mu\mapsto\mu_{e}$; moreover, it
is clear from the results in §III.10 (or Theorem 1.5, below, in the non-
weighted case) that $\mu\mapsto\mu_{e}$ is a weak∗-continuous isometric
algebra isomorphism of ${\cal M}_{r}(G)$ onto $M(G)$. Thus, for the purposes
of abstract harmonic analysis on (non-weighted) $G$, ${\cal M}_{r}(G)$ can be
used in place of the usual $M(G)$, and, as shown in [4], provides some
advantages.

For $\mu\in{\cal M}(G)$, let $I_{\mu}$ denote the linear functional
$I_{\mu}(f)=\int f\,d\mu$ defined on ${\cal L}^{1}(\mu)$, or any subspace of
${\cal L}^{1}(\mu)$. Then

$\displaystyle\mu\mapsto I_{\mu}:{\cal M}(G)\rightarrow{\mathfrak{I}}$ (3)

is a linear bijection where $\mathfrak{I}$ is the set of all linear
functionals $I$ on $C_{00}(G)$ such that $I\in C_{K}(G)^{*}$ for each
$K\in\mathfrak{C}(G)$; (3) maps ${\cal M}(G)^{+}$ onto ${\mathfrak{I}}^{+}$
and ${\cal M}_{r}(G)$ onto $C_{00}(G)^{*}=C_{0}(G)^{*}$ (II.8.12).

######  Remark 1.2.

It should be noted that when $\mu$ is a complex measure on a $\delta$-ring
$\mathfrak{S}$, $f\in{\cal L}^{1}(\mu)$ requires that $f$ vanish off a
countable union of sets in $\mathfrak{S}$ (II.2.5, paragraph 2). Thus, when
$f\in{\cal L}^{1}(\mu)$ for $\mu\in{\cal M}(G)$, $f$ must vanish off a
$\sigma$-compact set, a technical issue requiring careful attention throughout
this note. Consider the case when $\mu\in{\cal M}_{r}(G)$. Then any $\phi\in
C_{0}(G)$ vanishes off a $\sigma$-compact set and since $\phi$ is continuous
and bounded, it is easy to see that $\phi\in{\cal L}^{1}(\mu)$. Assuming
further that $\mu\geq 0$ and $\phi\geq 0$ and taking an increasing sequence
$(\phi_{n})$ in $C_{00}(G)^{+}$ such that
$\|\phi_{n}-\phi\|_{\infty}\rightarrow 0$, $\lim
I_{\mu}(\phi_{n})=\lim\int\phi_{n}\,d\mu=\int\phi\,d\mu=I_{\mu}(\phi)$ (e.g.,
by MCT II.7), so $I_{\mu}$ is the unique continuous extension of $I_{\mu}$ on
$C_{00}(G)$ to $C_{0}(G)$. Thus, $C_{0}(G)^{*}=\\{I_{\mu}:\mu\in{\cal
M}_{r}(G)\\}$, so — in this theory and as usual — we can identify ${\cal
M}_{r}(G)$ and $C_{0}(G)^{*}$ through the pairing
$\langle\mu,\phi\rangle=\int\phi\,d\mu$.

Let $\nu\in{\cal M}(G)$, $h$ a continuous function on $G$. Then $h$ is locally
$\nu$-measurable (II.8.2) and for each $A\in\mathfrak{S}(G)$,
$h\chi_{A}\in{\cal L}^{1}(\nu)$ since $|h|$ is bounded on $A$; i.e., $h$ is
locally $\nu$-summable. Therefore,

$h\nu(A):=\int h\chi_{A}\,d\nu\qquad(A\in\mathfrak{S}(G))$

defines a complex measure on $\mathfrak{S}(G)$ (see II.7.2, where the notation
$h\,d\nu$ rather than $h\nu$ is used); as $h\nu\ll\nu$ (II.7.8), $h\nu\in{\cal
M}(G)$ (II.8.3). If $h>0$, then ${1\over h}(h\nu)\in{\cal M}(G)$ and a simple
application of II.7.5 gives ${1\over h}(h\nu)=\nu$.

Hence, $\omega\nu\sim\nu$ for each $\nu\in{\cal M}(G)$, and

${\cal M}(G)\rightarrow{\cal M}(G):\nu\mapsto\omega\nu$

defines a linear isomorphism with inverse $\nu\mapsto{1\over\omega}\nu$. We
can thus define

${\cal M}(G,\omega):=\\{\nu\in{\cal M}(G):\omega\nu\in{\cal
M}_{r}(G)\\};\quad{\rm letting}\ \ \ \|\nu\|_{\omega}=\|\omega\nu\|\ \ \
(\nu\in{\cal M}(G,\omega)),$

it follows that ${\cal M}(G,\omega)$ is a Banach space and
$\nu\mapsto\omega\nu$ is an isometric linear isomorphism of ${\cal
M}(G,\omega)$ onto ${\cal M}_{r}(G)$ with inverse map
$\mu\mapsto{1\over\omega}\mu$. (As shown in the introduction, this definition
cannot, in general, be made with $M(G)$ replacing ${\cal M}_{r}(G)$.) Observe
that by II.7.3, $\nu\in{\cal M}(G,\omega)$ exactly when $|\nu|\in{\cal
M}(G,\omega)$, and $\|\nu\|_{\omega}=\||\nu|\|_{\omega}$.

######  Proposition 1.3.

For each $\nu\in{\cal M}(G,\omega)$, $I_{\nu}\in C_{0}(G,\omega^{-1})^{*}$ and
$\|I_{\nu}\|=\|\nu\|_{\omega}$; moreover,

$\displaystyle C_{0}(G,\omega^{-1})^{*}=\\{I_{\nu}:\nu\in{\cal
M}(G,\omega)\\}.$ (4)

We can thus make the identification ${\cal
M}(G,\omega)=C_{0}(G,\omega^{-1})^{*}$ through the pairing

$\langle\nu,\psi\rangle_{\omega}=\int\psi\,d\nu\qquad(\nu\in{\cal
M}(G,\omega),\psi\in C_{0}(G,\omega^{-1})).$

With respect to this identification, the inverse isometric isomorphisms

${\cal M}(G,\omega)\rightarrow{\cal M}_{r}(G):\nu\mapsto\omega\nu\quad{\rm
and}\quad{\cal M}_{r}(G)\rightarrow{\cal
M}(G,\omega):\mu\mapsto{1\over\omega}\mu$

are weak∗-homeomorphisms.

###### Proof.

As noted above, $S:C_{0}(G,\omega^{-1})\rightarrow
C_{0}(G):\psi\mapsto{\psi\over\omega}$ is an isometric isomorphism, so
$S^{*}:{\cal M}_{r}(G)=C_{0}(G)^{*}\rightarrow C_{0}(G,\omega^{-1})^{*}$ is
also an isometric isomorphism. Let $\nu\in{\cal M}(G,\omega)$. Then
$\omega\nu\in{\cal M}_{r}(G)$ and for $\psi\in C_{0}(G,\omega^{-1})$,
${\psi\over\omega}\in C_{0}(G)\subseteq{\cal L}^{1}(\omega\nu)$ (see Remark
1.2); therefore by II.7.5, $\psi=(\psi/\omega)\omega\in{\cal L}^{1}(\nu)$ and

$\langle
I_{\nu},\psi\rangle=\int\psi\,d\nu=\int{\psi\over\omega}\,d(\omega\nu)=\langle\omega\nu,S(\psi)\rangle=\langle
S^{*}(\omega\nu),\psi\rangle.$

Hence, $C_{0}(G,\omega^{-1})\subseteq{\cal L}^{1}(\nu)$,
$I_{\nu}=S^{*}(\omega\nu)\in C_{0}(G,\omega^{-1})^{*}$, and therefore
$\|I_{\nu}\|=\|S^{*}(\omega\nu)\|=\|\omega\nu\|=\|\nu\|_{\omega};$ since
$S^{*}(\mu)=I_{\omega^{-1}\mu}$ and $S^{*}$ maps onto
$C_{0}(G,\omega^{-1})^{*}$, we have (4). Making the identification of $\nu$
and $I_{\nu}$, $\mu\mapsto{1\over\omega}\mu=S^{*}(\mu)$ is weak∗-continuous,
with (weak∗-continuous) inverse map $\nu\mapsto\omega\nu$. ∎

In Lemma 1.4, $X$ is a locally compact Hausdorff space,
$h:X\rightarrow(0,\infty)$ is a continuous function, and $\mu\in{\cal
M}(X)^{+}$ is such that $h\mu\in{\cal M}_{r}(X)$. Observe that ${\cal
E}_{\mu}\subseteq{\mathfrak{B}}(X)={\cal E}_{h\mu}$; see Remark 1.1.

######  Lemma 1.4.

The function $h$ is locally $\mu_{e}$-summable and for any set $A\in{\cal
E}_{\mu}$, $h(\mu_{e})(A)=(h\mu)_{e}(A)$.

###### Proof.

Let $A\in{\cal E}_{\mu}$. Take $(C_{n})_{n}$ to be an increasing sequence of
compact subsets of $A$ such that $\mu_{e}(A)=\lim_{n}\mu(C_{n})$ and let
$D=\cup_{n}C_{n}$. Observe that $D,A\backslash D\in{\cal E}_{\mu}$ and
$\mu_{e}(D)=\lim\mu_{e}(C_{n})=\lim\mu(C_{n})=\mu_{e}(A)$; hence

$\displaystyle\mu_{e}(A\backslash D)=0.$ (5)

It follows that for any compact subset $C$ of $A\backslash D$, $\mu(C)=0$ and
therefore, since $h$ is locally $\mu$-summable and bounded on $C$,
$h\mu(C)=0$. Hence,

$\displaystyle\lim(h\mu)_{e}(A\backslash
D)=\lim\\{(h\mu)(C):C\in\mathfrak{C}(X),\ C\subseteq A\backslash D\\}=0.$ (6)

As noted in Remark 1.1, $h\chi_{A\backslash D}$ is locally
$\mu_{e}$-measurable and it follows from (5) and II.2.7 that

$\displaystyle\int h\chi_{A\backslash D}\,d\mu_{e}=\lim_{n}\int(h\wedge
n)\chi_{A\backslash D}\,d\mu_{e}=0.$ (7)

Also, since $h\mu$ is bounded, $\lim\int h\chi_{C_{n}}\,d\mu_{e}=\lim\int
h\chi_{C_{n}}\,d\mu=\sup(h\mu)(C_{n})<\infty$ (using II.8.15 Remark 3), and
therefore by II.2.7,

$\displaystyle\int h\chi_{D}\,d\mu_{e}=\lim\int
h\chi_{C_{n}}\,d\mu_{e}=\lim(h\mu)(C_{n})=\lim(h\mu)_{e}(C_{n})=(h\mu)_{e}(D).$
(8)

From (7) and (8), $h\chi_{A\backslash D},h\chi_{D}\in{\cal L}^{1}(\mu_{e})$,
whence $h\chi_{A}\in{\cal L}^{1}(\mu_{e})$. Hence, $h$ is locally $\mu_{e}$
summable. Moreover, (8), (7) and (6) yield $h(\mu_{e})(A)=(h\mu)_{e}(A)$. ∎

Let $p:G\times G\rightarrow G:(s,t)\mapsto st$. Following III.10.2, we say
that $\mu,\nu\in{\cal M}(G)$ are convolvable, or that $\mu*\nu$ exists, if $p$
is $\mu\times\nu$-proper in the sense of II.10.3, i.e., if $p^{-1}(A)\in{\cal
E}_{\mu\times\nu}$ whenever $A\in\mathfrak{S}(G)$. In this case,
$\mu*\nu\in{\cal M}(G)$, where for $A\in\mathfrak{S}(G)$,

$\displaystyle\mu*\nu(A)$ $\displaystyle=$ $\displaystyle
p_{*}((\mu\times\nu)_{e})(A)=(\mu\times\nu)_{e}(p^{-1}(A))$ $\displaystyle=$
$\displaystyle\lim\\{(\mu\times\nu)(C):C\subseteq p^{-1}(A),\
C\in\mathfrak{C}(G\times G)\\};$

see III.10.2, II.10.3, II.10.5, II.10.1. Equivalently, one can check that
$\mu*\nu$ exists if and only if

$\sup\\{(|\mu|\times|\nu|)(C):C\subseteq p^{-1}(D),\ C\in\mathfrak{C}(G\times
G)\\}<\infty$

for every compact subset $D$ of $G$. (In our context, the definition of
$\mu\times\nu\in{\cal M}(G\times G)$ and its properties are found in §II.9.)

######  Theorem 1.5.

With respect to convolution product, ${\cal
M}(G,\omega)=C_{0}(G,\omega^{-1})^{*}$ is a Banach algebra, i.e.,
$(\mu,\nu)\mapsto\mu*\nu$ is a well-defined associative operation on ${\cal
M}(G,\omega)$ satisfying
$\|\mu*\nu\|_{\omega}\leq\|\mu\|_{\omega}\|\nu\|_{\omega}$. Moreover, for
$\mu,\nu\in{\cal M}(G,\omega)$ and $\psi\in C_{0}(G,\omega^{-1})$,

$\displaystyle\langle\mu*\nu,\psi\rangle_{\omega}=\int\psi(st)\,d(\mu\times\nu)_{e}(s,t)=\iint\psi(st)\,d\mu(s)d\nu(t)=\iint\psi(st)\,d\nu(t)d\mu(s).$
(9)

###### Proof.

Let $\mu,\nu\in{\cal M}(G,\omega)$, with $\mu,\nu\geq 0$. Let $D$ be a compact
subset of $G$, $C$ a compact subset of $p^{-1}(D)$. The functions $1_{C}(x,y)$
and $g(x,y)={1\over\omega(x)\omega(y)}1_{C}(x,y)$ are Borel measurable
functions, and are therefore locally $(\sigma\times\rho)$-measurable for any
pair of measures $\sigma,\rho\in{\cal M}(G)$; moreover, since they are non-
negative, bounded and vanish off $C$, $1_{C},g\in{\cal
L}^{1}(\sigma\times\rho)$. Applying the Fubini Theorem (II.9.8) to these
functions, and using II.7.5 twice — which also applies by II.9.8 — we obtain

$\displaystyle\mu\times\nu(C)$ $\displaystyle=$ $\displaystyle\int\int
1_{C}(x,y)\,d\mu(x)d\nu(y)=\int\int
g(x,y)\omega(x)\,d\mu(x)\omega(y)\,d\nu(y)$ $\displaystyle=$
$\displaystyle\int\int g(x,y)\,d\omega\mu(x)\,d\omega\nu(y)=\int_{G\times
G}{1\over\omega(x)\omega(y)}1_{C}(x,y)\,d(\omega\mu\times\omega\nu)(x,y)$
$\displaystyle\leq$ $\displaystyle\int_{G\times
G}{1\over\omega(xy)}1_{C}(x,y)\,d(\omega\mu\times\omega\nu)(x,y)\leq\int_{G\times
G}M_{D}1_{C}(x,y)\,d(\omega\mu\times\omega\nu)_{e}(x,y)$

where $M_{D}=\sup_{z\in D}\omega(z)^{-1}$, since $C\subseteq p^{-1}(D)$, and
we have used II.8.15 Remark 3. Observe that
$p^{-1}(D)\in{\mathfrak{B}}(G\times G)={\cal E}_{\omega\mu\times\omega\nu}$,
since $\omega\mu\times\omega\nu\in{\cal M}_{r}(G\times G)$ — see II.9.14 — so

$\displaystyle\mu\times\nu(C)\leq\int_{G\times
G}M_{D}1_{p^{-1}(D)}\,d(\omega\mu\times\omega\nu)_{e}\leq
M_{D}\|\omega\mu\times\omega\nu\|=M_{D}\|\omega\mu\|\|\omega\nu\|=M_{D}\|\mu\|_{\omega}\|\nu\|_{\omega}.$

Hence, $\mu*\nu$ exists. We now show $\mu*\nu\in{\cal M}(G,\omega)$ and
$\|\mu*\nu\|_{\omega}\leq\|\mu\|_{\omega}\|\nu\|_{\omega}$. Let
$A\in\mathfrak{S}(G)$. Since $\omega$ is continuous on $G$ and
$\mu*\nu\in{\cal M}(G)$, $\omega$ is locally $\mu*\nu$-summable and
$\omega(\mu*\nu)\in{\cal M}(G)$. Hence, $\omega\chi_{A}\in{\cal
L}^{1}(\mu*\nu)={\cal L}^{1}(p_{*}(\mu\times\nu)_{e})$. Therefore, II.10.2
gives $(\omega\chi_{A})\circ p\in{\cal L}^{1}((\mu\times\nu)_{e})$ and

$\displaystyle\omega(\mu*\nu)(A)$ $\displaystyle=$
$\displaystyle\int\omega\chi_{A}\,d(p_{*}((\mu\times\nu)_{e}))=\int(\omega\chi_{A})\circ
p\,d(\mu\times\nu)_{e}$ $\displaystyle=$ $\displaystyle\int\omega\circ
p\,\chi_{p^{-1}(A)}\,d(\mu\times\nu)_{e}\leq\int(\omega\times\omega)\chi_{p^{-1}(A)}\,d(\mu\times\nu)_{e}$

where $(\omega\times\omega)(s,t)=\omega(s)\omega(t)$. By II.9.9 and II.9.3,
$(\omega\times\omega)(\mu\times\nu)=\omega\mu\times\omega\nu$, which belongs
to ${\cal M}_{r}(G\times G)$ by II.9.14. Observe that $\omega\times\omega$ is
locally $(\mu\times\nu)_{e}$-summable, by Lemma 1.4, and $p^{-1}(A)\in{\cal
E}_{\mu\times\nu}$, since $\mu*\nu$ exists. Hence, the above inequality and
Lemma 1.4 yield

$\displaystyle\omega(\mu*\nu)(A)$ $\displaystyle\leq$
$\displaystyle(\omega\times\omega)(\mu\times\nu)_{e}(p^{-1}(A))=((\omega\times\omega)(\mu\times\nu))_{e}(p^{-1}(A))$
$\displaystyle=$
$\displaystyle(\omega\mu\times\omega\nu)_{e}(p^{-1}(A))\leq\|\omega\mu\times\omega\nu\|=\|\omega\mu\|\|\omega\nu\|=\|\mu\|_{\omega}\|\nu\|_{\omega}.$

Hence, $\omega(\mu*\nu)$ is bounded, i.e., $\mu*\nu\in{\cal M}(G,\omega)$, and
$\|\mu*\nu\|_{\omega}=\|\omega(\mu*\nu)\|\leq\|\mu\|_{\omega}\|\nu\|_{\omega}.$

Assume now that $\mu,\nu$ are any two measures in ${\cal M}(G,\omega)$. As we
have noted, $\sigma\in{\cal M}(G,\omega)$ exactly when $|\sigma|\in{\cal
M}(G,\omega)$ and $\|\sigma\|_{\omega}=\||\sigma|\|_{\omega}$, so it follows
from III.10.3 and the positive case that $\mu*\nu$ exists and
$|\mu*\nu|\leq|\mu|*|\nu|$. Hence, $\omega|\mu*\nu|\leq\omega|\mu|*|\nu|$, so
$\mu*\nu\in{\cal M}(G,\omega)$ and

$\|\mu*\nu\|_{\omega}=\|\omega|\mu*\nu|\|\leq\|\omega|\mu|*|\nu|\|=\||\mu|*|\nu|\|_{\omega}\leq\||\mu|\|_{\omega}\||\nu|\|_{\omega}=\|\mu\|_{\omega}\|\nu\|_{\omega}.$

Associativity of convolution in ${\cal M}(G,\omega)$ is now an immediate
consequence of III.10.10. Since any $\psi\in C_{0}(G,\omega^{-1})$ vanishes
off a $\sigma$-compact subset of $G$ and any $\mu,\nu\in{\cal M}(G,\omega)$
are $\sigma$-bounded — since $\omega\mu$ and $\omega\nu$ are so, and
$\omega\mu\sim\mu$, $\omega\nu\sim\nu$ — Remark III.10.8 applies to give (9).
∎

Let $\lambda=\lambda_{G}$ be a fixed left Haar measure on $G$, ${\cal
L}^{1}(G)={\cal L}^{1}(\lambda)$. Then $\lambda\in{\cal M}(G)$ (§III.7), so
$\omega\lambda\in{\cal M}(G)$ as well and, since $\omega>0$,
$\omega\lambda\sim\lambda$, from which it follows that $g$ is locally
$\omega\lambda$-measurable and vanishes off a $\sigma$-compact set if and only
if $g\omega$ is locally $\lambda$-measurable and vanishes off a
$\sigma$-compact set. Hence, if we define ${\cal L}^{1}(G,\omega):={\cal
L}^{1}(\omega\lambda)$, $g\in{\cal L}^{1}(G,\omega)$ exactly when
$g\omega\in{\cal L}^{1}(G)$, and in this case $\int g\,d(\omega\lambda)=\int
g\omega\,d\lambda$, by II.7.5. Thus,

${\cal L}^{1}(G,\omega)=\\{g:g\omega\in{\cal L}^{1}(G)\\}\ \ {\rm and}\ \
\|g\|_{\omega}:=\|g\|_{{\cal L}^{1}(\omega\lambda)}=\|g\omega\|_{1}$

defines a Banach space norm on ${\cal L}^{1}(G,\omega)$. Moreover, $T:{\cal
L}^{1}(G,\omega)\rightarrow{\cal L}^{1}(G):g\mapsto g\omega$ is an isometric
linear isomorphism, with inverse $f\mapsto{1\over\omega}f$, so $T^{*}:{\cal
L}^{\infty}(G)={\cal L}^{1}(G)^{*}\rightarrow{\cal L}^{1}(G,\omega)^{*}$ is a
weak∗-continuous isometric isomorphism given by $\langle
T^{*}\phi,g\rangle_{\omega}=\langle\phi,\omega
g\rangle=\int(\phi\omega)g\,d\lambda.$ Letting

${\cal L}^{\infty}(G,\omega^{-1}):=\\{\phi\omega:\phi\in{\cal
L}^{\infty}(G)\\}=\left\\{\psi:{\psi\over\omega}\in{\cal
L}^{\infty}(G)\right\\}\ \ \ {\rm where}\ \ \
\|\psi\|_{\infty,\omega^{-1}}:=\left\|{\psi\over\omega}\right\|_{\infty},$

we can hence identify ${\cal L}^{1}(G,\omega)^{*}$ with ${\cal
L}^{\infty}(G,\omega^{-1})$ via the pairing
$\langle\psi,g\rangle_{\omega}=\int\psi g\,d\lambda.$ Observe that
$S=(T^{*})^{-1}:{\cal L}^{\infty}(G,\omega^{-1})\rightarrow{\cal
L}^{\infty}(G):\psi\mapsto{\psi\over\omega}$ is a weak∗-homeomorphic isometric
isomorphism. (We note that ${\cal L}^{\infty}(G,\omega^{-1})$ is not usually
the same space as ${\cal L}^{\infty}(\omega\lambda)$ ($={\cal
L}^{\infty}(\lambda)$ because $\omega\lambda\sim\lambda$), which can also be
identified with ${\cal L}^{1}(\omega\lambda)^{*}={\cal L}^{1}(G,\omega)^{*}$
in the usual way by II.7.11.) Note that because $T^{-1}$ maps $C_{00}(G)$ onto
itself, $C_{00}(G)$ is dense in ${\cal L}^{1}(G,\omega)$.

Let $g\in{\cal L}^{1}(G,\omega)={\cal L}^{1}(\omega\lambda)$,
$A\in\mathfrak{S}(G)$. Then $\omega g\in{\cal L}^{1}(\lambda)$ and
${1\over\omega}$ is bounded on $A$, so
$\chi_{A}g=({1\over\omega}\chi_{A})\omega g\in{\cal L}^{1}(\lambda)$; hence,
$g\lambda\in{\cal M}(G)$ is well-defined (II.7.2). Also,
$\omega(g\lambda)=(\omega g)\lambda\in{\cal M}(G)$ by II.7.5 and, by
II.7.9/III.11.3, $\|f\|_{1}=\|f\lambda\|$ for $f\in{\cal L}^{1}(G)$ and

${\cal M}_{a}(G)=\\{\mu\in{\cal
M}_{r}(G):\mu\ll\lambda\\}=\\{f\lambda:f\in{\cal L}^{1}(G)\\}=\\{(\omega
g)\lambda:g\in{\cal L}^{1}(G,\omega)\\}.$

Since $\omega\nu\sim\nu$ for any $\nu\in{\cal M}(G)$, it readily follows that
$g\mapsto g\lambda:{\cal L}^{1}(G,\omega)\rightarrow{\cal M}_{a}(G,\omega)$ is
a surjective linear isometry, where ${\cal M}_{a}(G,\omega):=\\{\nu\in{\cal
M}(G,\omega):\nu\ll\lambda\\}.$ We can thus identify ${\cal L}^{1}(G,\omega)$
with ${\cal M}_{a}(G,\omega)$ via $g\mapsto g\lambda$.

######  Proposition 1.6.

The Banach space ${\cal L}^{1}(G,\omega)={\cal M}_{a}(G,\omega)$ is a closed
ideal in ${\cal M}(G,\omega)$ and has a positive contractive approximate
identity. Moreover, if $g\in{\cal L}^{1}(G,\omega)$ and $\nu\in{\cal
M}(G,\omega)$, then $\nu*g,g*\nu\in{\cal L}^{1}(G,\omega)$ are given by the
formulas, which hold for locally $\lambda$-almost all $t\in G$,

$\displaystyle\nu*g(t)=\int g(s^{-1}t)\,d\nu(s)\quad{\rm and}\quad
g*\nu(t)=\int\Delta(s^{-1})g(ts^{-1})\,d\nu(s);$ (10)

thus, ${\cal L}^{1}(G,\omega)$ is a Banach algebra with respect to the
convolution product

$\displaystyle f*g(t)=\int f(s)g(s^{-1}t)\,d\lambda(s).$ (11)

###### Proof.

We have already noted that $g$ is locally $\lambda$-summable and vanishes off
a $\sigma$-compact set, and $(g\lambda)*\nu$, $\nu*(g\lambda)$ exist in ${\cal
M}(G,\omega)$ by Theorem 1.5. Letting $h(t)$ and $k(t)$ be defined by the
respective integral formulas on the left and right of (10),
$\nu*(g\lambda)=h\lambda$ and $(g\lambda)*\nu=k\lambda$ by III.11.5. Thus,
$h\lambda,k\lambda\in{\cal M}_{a}(G,\omega)=\\{f\lambda:f\in{\cal
L}^{1}(G,\omega)\\}$, so the uniqueness part of the Radon–Nikodym Theorem —
see Remark 1 of II.7.8 — implies that $h,k\in{\cal L}^{1}(G,\omega)$. The
formula (11) now follows quickly (or directly from III.11.6). Let $\cal I$ be
the neighbourhood system at $e_{G}$ and for each $\alpha\in{\cal I}$, let
$f_{\alpha}\in C_{00}(G)$ be chosen with $f_{\alpha}\geq 0$,
$\|f_{\alpha}\|_{1}=1$ and support contained in $\alpha$. Then
$(f_{\alpha})_{\alpha}$ is a bounded approximate identity for ${\cal
L}^{1}(G)$. Letting $e_{\alpha}=\omega^{-1}f_{\alpha}$,
$\|e_{\alpha}\|_{\omega}=1$ and $\|e_{\alpha}\|_{1}\rightarrow 1$, from which
it easily follows that $(e_{\alpha})_{\alpha}$ is also a bounded approximate
identity for ${\cal L}^{1}(G)$; the proof of Lemma 2.1 in [5] now shows that
$(e_{\alpha})_{\alpha}$ is a contractive approximate identity for ${\cal
L}^{1}(G,\omega)$. ∎

######  Remark 1.7.

Every Borel measurable function is locally $\lambda$-measurable and every
$f\in L^{1}(G,\omega)$ — where $L^{1}(G,\omega)$ is defined in the usual sense
(as in the introduction) — vanishes off a $\sigma$-compact set. It follows
that the Banach algebra ${\cal L}^{1}(G,\omega)$, as we have defined it,
exactly coincides with the usual definition of the Beurling group algebra
$L^{1}(G,\omega)$, which, as noted in the introduction, is always valid. Going
forward, we can therefore use any known result about $L^{1}(G,\omega)={\cal
L}^{1}(G,\omega)$ that was proved independently of $M(G,\omega)$.

##  2 The dual Banach algebra ${\cal M}(G,\omega)$ and the embedding map

The support of $\mu$ in ${\cal M}(G)$ is the set ${\rm
s}(\mu)=G\backslash\bigcup\\{U\in\mathfrak{S}(G):U\ {\rm is\ open\ and\
}|\mu|(U)=0\\}$(II.8.9). Let ${\cal M}_{cr}(G)=\\{\mu\in{\cal M}(G):{\rm
s}(\mu){\rm\ is\ compact}\\}.$

######  Remark 2.1.

1\. Observe that ${\rm s}(\mu)={\rm s}(\mu_{e})=G\backslash\bigcup\\{V\in{\cal
E}_{\mu}:V\ {\rm is\ open\ and\ }|\mu_{e}|(V)=0\\}$.

2\. Since $\omega$ and ${1\over\omega}$ are bounded on any set $A$ in
$\mathfrak{S}(G)$, ${\rm s}(\mu)={\rm s}(\omega\mu)={\rm
s}\left({1\over\omega}\mu\right)$ for any $\mu\in{\cal M}(G)$.

3\. By III.10.16, ${\cal M}_{cr}(G)$ is a dense subalgebra of ${\cal
M}_{r}(G)$. From 2 above, the inverse linear isometries $\nu\mapsto\omega\nu$
and $\mu\mapsto{1\over\omega}\mu$ between ${\cal M}(G,\omega)$ and ${\cal
M}_{r}(G)$ map ${\cal M}_{cr}(G)$ onto itself, so ${\cal M}_{cr}(G)$ is also a
dense subalgebra of ${\cal M}(G,\omega)$.

A measure $\sigma$ on a $\delta$-ring $\mathfrak{S}$ is concentrated on a set
$F$ if for each $A\in\mathfrak{S}$, $A\cap F,A\backslash F\in\mathfrak{S}$ and
$\sigma(A)=\sigma(A\cap F)$ or, equivalently, $\sigma(A\backslash F)=0$. For
$\mu\in{\cal M}(G)$ and a Borel set $F$, $A\cap F,A\backslash
F\in\mathfrak{S}(G)$ (respectively, $A\cap F,A\backslash F\in{\cal E}_{\mu}$)
is automatic for any $A\in\mathfrak{S}(G)$ ($A\in{\cal E}_{\mu}$), and it is
clear from (2) that $\mu$ is concentrated on $F$ if and only if $\mu_{e}$ is
concentrated on $F$. A function $\psi\in LUC(G,\omega^{-1})$ may fail to
vanish off a $\sigma$-compact set and therefore, as noted in Remark 1.2, in
this theory we cannot integrate $\psi$ with respect to any $\mu$ in ${\cal
M}(G)$. Lemma 2.2 allows us to move past this issue.

######  Lemma 2.2.

(a) Every $\mu$ in ${\cal M}(G)$ is concentrated on its support, ${\rm
s}(\mu)$.
(b) Let $\mu\in{\cal M}_{r}(G)$. Then $\mu$ (and therefore $\mu_{e}$) is
concentrated on a $\sigma$-compact subset $F$ of $G$ and, for any such $F$ and
any Borel measurable function $f\in{\cal L}^{1}(\mu_{e})$, $f\chi_{F}\in{\cal
L}^{1}(\mu)\cap{\cal L}^{1}(\mu_{e})$ and

$\int f\,d\mu_{e}=\int f\chi_{F}\,d\mu_{e}=\int f\chi_{F}\,d\mu.$

(c) Any $\nu\in{\cal M}(G,\omega)$ is concentrated on a $\sigma$-compact set.

###### Proof.

(a) Let $A\in\mathfrak{S}(G)$. Any compact subset of $A\backslash{\rm s}(\mu)$
is covered by the collection of open sets $U\in\mathfrak{S}(G)$ with
$|\mu|(U)=0$, and is therefore $|\mu|$-null; by regularity of $\mu$
(II.8.2(II)), $|\mu|(A\backslash{\rm s}(\mu))=0$.

(b) Take $(C_{n})_{n}$ to be an increasing sequence of compact subsets of
${\rm s}(\mu)$ such that $|\mu|(C_{n})>\|\mu\|-1/n$ and let $F=\bigcup C_{n}$,
where we have used (b). Then $\mu$ is concentrated on $F$ because for
$A\in\mathfrak{S}(G)$,

$|\mu|(A\backslash F)=|\mu|((A\backslash F)\cap{\rm s}(\mu))\leq|\mu|_{e}({\rm
s}(\mu)\backslash F)=|\mu|_{e}({\rm
s}(\mu))-|\mu_{e}|(F)=\|\mu\|-\lim|\mu|(C_{n})=0.$

Suppose $\mu\geq 0$, $F$ is any $\sigma$-compact set on which $\mu$ is
concentrated, and $f\in{\cal L}^{1}(\mu_{e})$ is a non-negative Borel-
measurable function. It is then clear (from II.2.2 and II.2.5) that
$f\chi_{F}\in{\cal L}^{1}(\mu_{e})$ and $\int f\,d\mu_{e}=\int
f\chi_{F}\,d\mu_{e}$. Also, $f\chi_{F}$ is locally $\mu$-measurable (II.8.2),
vanishes off the $\sigma$-compact set $F$ and, taking any sequence of non-
negative $\mathfrak{S}(G)$-simple functions such that $h_{n}\uparrow
f\chi_{F}$, II.2.2 gives

$\int f\chi_{f}\,d\mu_{e}=\lim\int h_{n}\,d\mu_{e}=\lim\int h_{n}\,d\mu=\int
f\chi_{F}\,d\mu.$

(c) Since $\omega\nu\in{\cal M}_{r}(G)$ and $\nu\sim\omega\nu$, this follows
from (b). ∎

Since ${\cal L}^{1}(G,\omega)$ is a closed ideal in ${\cal M}(G,\omega)$,
through the operations

$\langle\psi\cdot\nu,g\rangle=\langle\psi,\nu*g\rangle\ \ {\rm and}\ \
\langle\nu\cdot\psi,g\rangle=\langle\psi,g*\nu\rangle\quad(\psi\in{\cal
L}^{\infty}(G,\omega^{-1}),\nu\in{\cal M}(G,\omega),g\in{\cal
L}^{1}(G,\omega)),$

${\cal L}^{\infty}(G,\omega^{-1})={\cal L}^{1}(G,\omega)^{*}$ is a dual ${\cal
M}(G,\omega)$-module. Observe that for $\psi\in{\cal
L}^{\infty}(G,\omega^{-1})$ and $s\in G$,

$\psi\cdot\delta_{s}(t)=\psi\cdot s(t):=\psi(st)\ \ {\rm and}\ \
\delta_{s}\cdot\psi(t)=s\cdot\psi(t):=\psi(ts)\quad(t\in G).$

Recall that $\psi$ belongs to $LUC(G,\omega^{-1})$ [$RUC(G,\omega^{-1})$] when
${\psi\over\omega}$ belongs to $LUC(G)$ [$RUC(G)$]. For $LUC(G,\omega^{-1})$,
the following is [9, Proposition 1.3] and [3, Propositions 7.15 and 7.17],
(where no restrictions are needed on the weight $\omega$); symmetric arguments
establish the $RUC(G,\omega^{-1})$ case.

######  Lemma 2.3.

The following statements are equivalent:

  * (a)

$\psi\in LUC(G,\omega^{-1})$ [$RUC(G,\omega^{-1})$];

  * (b)

$\psi\in\ell^{\infty}(G,\omega^{-1})$ and the map
$G\rightarrow(\ell^{\infty}(G,\omega^{-1}),\|\cdot\|_{\infty,\omega^{-1}}):s\mapsto\psi\cdot
s\ [s\cdot\psi]$ is continuous;

  * (c)

$\psi\in{\cal L}^{\infty}(G,\omega^{-1})$ and the map $G\rightarrow({\cal
L}^{\infty}(G,\omega^{-1}),\|\cdot\|_{\infty,\omega^{-1}}):s\mapsto\psi\cdot
s\ [s\cdot\psi]$ is continuous;

  * (d)

$\psi\in{\cal L}^{\infty}(G,\omega^{-1})\cdot{\cal L}^{1}(G,\omega)$
[$\psi\in{\cal L}^{1}(G,\omega)\cdot{\cal L}^{\infty}(G,\omega^{-1})$].

######  Remark 2.4.

1\. Observe that condition (b) implies $\psi$ is continuous on $G$, whence
$\psi\in{\cal L}^{\infty}(G,\omega^{-1})$.

2\. In the proof of [3, Proposition 7.15], the authors establish continuity of
a function $\psi$ satisfying (c) via Ascoli’s theorem. An alternative approach
is to establish (i) and (ii) as follows:
(i) If $\phi\in{\cal L}^{\infty}(G,\omega^{-1})$ and $g\in{\cal
L}^{1}(G,\omega)$, then $\phi\cdot g$ can be identified with the continuous
function

$\displaystyle(\phi\cdot g)(t)=\langle\phi,g*\delta_{t}\rangle\qquad{\rm for\
every}\ t\in G.$ (12)

[Note that $H\in\ell^{\infty}(G,\omega^{-1})$ where
$H(t):=\langle\phi,g*\delta_{t}\rangle$ and, since $t\mapsto
g*\delta_{t}:G\rightarrow({\cal L}^{1}(G,\omega),\|\cdot\|_{\omega})$ is
continuous — e.g., see [19, Lemma 3.1.5], which holds for any weight $\omega$
— $H$ is continuous on $G$ (and satisfies Lemma 2.3(c)); in a standard way,
one can check that for $f\in{\cal L}^{1}(G,\omega)$, $\langle\phi\cdot
g,f\rangle=\langle H,f\rangle$.]
(ii) If $\psi$ satisfies (c) and $(e_{i})$ is a bounded approximate identity
for ${\cal L}^{1}(G,\omega)$, then $\|\psi\cdot
e_{i}-\psi\|_{\infty,\omega^{-1}}\rightarrow 0$; since $CB(G,\omega^{-1})$ is
closed in ${\cal L}^{\infty}(G,\omega^{-1})$, $\psi\in CB(G,\omega^{-1})$.

######  Proposition 2.5.

The spaces $LUC(G,\omega^{-1})$ and $RUC(G,\omega^{-1})$ are ${\cal
M}(G,\omega)$-submodules of ${\cal L}^{\infty}(G,\omega^{-1})$. Moreover, for
$\nu\in{\cal M}(G,\omega)$, $\psi\in LUC(G,\omega^{-1})$ [$\psi\in
RUC(G,\omega^{-1})$] and for every $s\in G$,

$(\nu\cdot\psi)(s)=\int(\psi\cdot s)\chi_{F_{s}}\,d\nu=\int{\psi\cdot
s\over\omega}\,\chi_{F_{s}}\,d(\omega\nu)=\int{\psi\cdot
s\over\omega}\,d(\omega\nu)_{e}$
$\left[(\psi\cdot\nu)(s)=\int(s\cdot\psi)\chi_{F_{s}}\,d\nu=\int{s\cdot\psi\over\omega}\,\chi_{F_{s}}\,d(\omega\nu)=\int{s\cdot\psi\over\omega}\,d(\omega\nu)_{e}\right],$

where $F_{s}$ is any $\sigma$-compact set on which $\nu$ is concentrated;
$F_{s}$ can be chosen to vary with $s\in G$.

###### Proof.

Letting $\nu\in{\cal M}(G,\omega)$, $\psi\in LUC(G,\omega^{-1})$, it is clear
from Lemma 2.3 (d) that $\psi\cdot\nu,\nu\cdot\psi\in LUC(G,\omega^{-1})$.
Since ${\psi\cdot s\over\omega}\in LUC(G)$ and $\omega\nu\in{\cal M}_{r}(G)$,

$H(s)=H_{\nu,\psi}(s):=\int{\psi\cdot
s\over\omega}\,d(\omega\nu)_{e}=\int{\psi\cdot
s\over\omega}\,\chi_{F_{s}}\,d(\omega\nu)$

is well-defined, where we have used Lemma 2.2. The function $(\psi\cdot
s)\chi_{F_{s}}\in\ell^{\infty}(G,\omega^{-1})$ is Borel measurable — and
therefore locally $\nu$-measurable — and vanishes off the $\sigma$-compact set
$F_{s}$, so ${\psi\cdot s\over\omega}\chi_{F_{s}}\in{\cal L}^{1}(\omega\nu)$.
Therefore, by II.7.5, $(\psi\cdot s)\chi_{F_{s}}\in{\cal L}^{1}(\nu)$ and

$\int(\psi\cdot s)\chi_{F_{s}}\,d\nu=\int{\psi\cdot
s\over\omega}\,\chi_{F_{s}}\,\omega\,d\nu=\int{\psi\cdot
s\over\omega}\,\chi_{F_{s}}\,d(\omega\nu)=H(s).$

Since $\displaystyle|H(s)|\leq\left\|{\psi\cdot
s\over\omega}\right\|_{\infty}\|\omega\nu\|\leq\omega(s)\|\psi\|_{\infty,\omega^{-1}}\|\nu\|_{\omega}$,
$H=H_{\nu,\psi}\in\ell^{\infty}(G,\omega^{-1})$ with
$\|H_{\nu,\psi}\|_{\infty,\omega^{-1}}\leq\|\psi\|_{\infty,\omega^{-1}}\|\nu\|_{\omega}$.
Hence, if $s_{i}\rightarrow s$ in $G$,

$\|(H_{\nu,\psi})\cdot s_{i}-(H_{\nu,\psi})\cdot
s\|_{\infty,\omega^{-1}}=\|H_{\nu,\psi\cdot s_{i}-\psi\cdot
s}\|_{\infty,\omega^{-1}}\leq\|\psi\cdot s_{i}-\psi\cdot
s\|_{\infty,\omega^{-1}}\|\nu\|_{\omega}\rightarrow 0;$

by Lemma 2.3, $H_{\nu,\psi}\in LUC(G,\omega^{-1})$. To show that
$H_{\nu,\psi}=\nu\cdot\psi$, we can assume $\nu\geq 0$, $\psi\geq 0$ and take
$F=F_{s}$ for each $s\in G$. Let $g\geq 0$ be a function in the dense subspace
$C_{00}(G)$ of ${\cal L}^{1}(G,\omega)$. Since the maps
$(s,t)\mapsto\psi(t)\Delta(s^{-1})g(ts^{-1})\chi_{F}(s),\
\psi(ts)g(t)\chi_{F}(s)$ are Borel measurable — hence locally
$(\nu\times\lambda)$-measurable — and vanish off a $\sigma$-compact subset of
$G\times G$, our applications of the Fubini Theorem (II.9.8) are valid in the
following calculation. Using (10):

$\displaystyle\langle\nu\cdot\psi,g\rangle$ $\displaystyle=$
$\displaystyle\langle\psi,g*\nu\rangle=\int\psi(t)\int\Delta(s^{-1})g(ts^{-1})\,d\nu(s)\,d\lambda(t)$
$\displaystyle=$
$\displaystyle\int\int\psi(t)\Delta(s^{-1})g(ts^{-1})\chi_{F}(s)\,d\nu(s)\,d\lambda(t)=\int\int\psi(t)\Delta(s^{-1})g(ts^{-1})\chi_{F}(s)\,d\lambda(t)\,d\nu(s)$
$\displaystyle=$
$\displaystyle\int\int\psi(ts)g(t)\chi_{F}(s)\,d\lambda(t)\,d\nu(s)=\int\int\psi\cdot
t(s)\chi_{F}(s)\,d\nu(s)\,g(t)\,d\lambda(t)=\langle H_{\nu,\lambda},g\rangle;$

since both functions are continuous, $\nu\cdot\lambda=H_{\nu,\lambda}$. ∎

######  Corollary 2.6.

The space $C_{0}(G,\omega^{-1})$ is a ${\cal M}(G,\omega)$-submodule of ${\cal
L}^{\infty}(G,\omega^{-1})$, and for $\nu\in{\cal M}(G,\omega)$, $\psi\in
C_{0}(G,\omega^{-1})$ and $s\in G$,

$\displaystyle\nu\cdot\psi(s)=\int\psi\cdot s\,d\nu=\langle\nu,\psi\cdot
s\rangle_{\omega}\ \ {\rm and}\ \ \psi\cdot\nu(s)=\int
s\cdot\psi\,d\nu=\langle\nu,s\cdot\psi\rangle_{\omega}.$ (13)

###### Proof.

Let $\psi\in C_{0}(G,\omega^{-1})$ and let $F$ be a $\sigma$-compact set on
which $\nu$ is concentrated. Taking $A_{s}$ to be a $\sigma$-compact set off
of which $\psi\cdot s$ and $s\cdot\psi$ vanish, and putting $F_{s}=F\cup
A_{s}$, Proposition 2.5 gives $\nu\cdot\psi,\psi\cdot\nu\in(LUC\cap
RUC)(G,\omega^{-1})$ and

$\nu\cdot\psi(s)=\int(\psi\cdot s)\chi_{F_{s}}\,d\nu=\int\psi\cdot s\,d\nu\ \
{\rm and}\ \ \psi\cdot\nu(s)=\int(s\cdot\psi)\chi_{F_{s}}\,d\nu=\int
s\cdot\psi\,d\nu.$

Observe that $\nu\cdot\psi$ is supported on ${\rm s}(\psi){\rm s}(\nu)^{-1}$,
which is compact when $\nu$ belongs to the dense subspace ${\cal M}_{cr}(G)$
of ${\cal M}(G,\omega)$ and $\psi$ belongs to the dense subspace $C_{00}(G)$
of $C_{0}(G,\omega^{-1})$. It follows that $C_{0}(G,\omega^{-1})$ is a left
(and similarly, right) ${\cal M}(G,\omega)$-submodule of ${\cal
L}^{\infty}(G,\omega^{-1})$. ∎

It follows that ${\cal M}(G,\omega)=C_{0}(G,\omega^{-1})^{*}$ is a dual ${\cal
M}(G,\omega)$-module with respect to the operations

$\langle\mu\cdot_{r}\nu,\psi\rangle_{\omega}=\langle\mu,\nu\cdot\psi\rangle_{\omega}\quad{\rm
and}\quad\langle\mu\cdot_{l}\nu,\psi\rangle_{\omega}=\langle\nu,\psi\cdot\mu\rangle_{\omega}\qquad(\mu,\nu\in{\cal
M}(G,\omega),\psi\in C_{0}(G,\omega^{-1})).$

However, from (9) and (13),

$\displaystyle\mu\cdot_{r}\nu=\mu*\nu=\mu\cdot_{l}\nu,$ (14)

so $(\mu,\nu)\mapsto\mu*\nu$ is separately weak∗-continuous on ${\cal
M}(G,\omega)$. Hence:

######  Corollary 2.7.

The Beurling measure algebra ${\cal M}(G,\omega)$ is a dual Banach algebra.

Let $A$ be a Banach algebra. Recall that a closed submodule ${\cal S}(A^{*})$
of the dual $A$-bimodule $A^{*}$ is left [right] introverted if for each
$\mu\in{\cal S}(A^{*})^{*}$ and $\phi\in{\cal S}(A^{*})$,
$\mu{\scriptscriptstyle\square}\phi\in{\cal S}(A^{*})$
[$\phi\diamond\mu\in{\cal S}(A^{*})$] where
$\mu{\scriptscriptstyle\square}\phi,\phi\diamond\mu\in A^{*}$ are defined by

$\langle\mu{\scriptscriptstyle\square}\phi,a\rangle_{A^{*}-A}=\langle\mu,\phi\cdot
a\rangle_{{\cal S}^{*}-{\cal S}}\quad{\rm
and}\quad\langle\phi\diamond\mu,a\rangle_{A^{*}-A}=\langle\mu,a\cdot\phi\rangle_{{\cal
S}^{*}-{\cal S}};$

in this case, ${\cal S}(A^{*})^{*}$ is a Banach algebra with respect to its
left [right] Arens product

$\langle\mu{\scriptscriptstyle\square}\nu,\phi\rangle=\langle\mu,\nu{\scriptscriptstyle\square}\phi\rangle\quad[\langle\mu\diamond\nu,\phi\rangle=\langle\nu,\phi\diamond\mu\rangle]\qquad(\mu,\nu\in{\cal
S}(A^{*})^{*},\ \phi\in{\cal S}(A^{*})).$

The map $\eta_{\cal S}:A\rightarrow{\cal S}(A^{*})^{*}$ defined by
$\langle\eta_{\cal S}(a),\phi\rangle=\langle\phi,a\rangle$ is a bounded
homomorphism with weak∗-dense range and, when $A$ is left introverted,
$\eta_{\cal S}$ maps into the topological centre of $({\cal
S}(A^{*})^{*},{\scriptscriptstyle\square})$, $Z_{t}({\cal
S}(A^{*})^{*})=\\{\mu\in{\cal
S}(A^{*})^{*}:\nu\mapsto\mu{\scriptscriptstyle\square}\nu\ {\rm is\
wk}^{*}-{\rm wk}^{*}\ {\rm continuous\ on\ }{\cal S}(A^{*})^{*}\\}.$ For this
see, e.g., [3].

######  Proposition 2.8.

The subspace $C_{0}(G,\omega^{-1})$ of ${\cal L}^{\infty}(G,\omega^{-1})={\cal
L}^{1}(G,\omega)^{*}$ is left and right introverted and
$\mu*\nu=\mu{\scriptscriptstyle\square}\nu=\mu\diamond\nu$ for
$\mu,\nu\in{\cal M}(G,\omega)=C_{0}(G,\omega^{-1})^{*}$.

###### Proof.

By Corollary 2.6, $C_{0}(G,\omega^{-1})$ is a ${\cal
L}^{1}(G,\omega)$-submodule of ${\cal L}^{\infty}(G,\omega^{-1})$. Let
$\mu,\nu\in{\cal M}(G,\omega)$, $\psi\in C_{0}(G,\omega^{-1})$. For $g\in{\cal
L}^{1}(G,\omega)$, equation (14) gives

$\displaystyle\langle\nu{\scriptscriptstyle\square}\psi,g\rangle_{{\cal
L}^{\infty}-{\cal L}^{1}}=\langle\nu,\psi\cdot g\rangle_{\omega}=\langle
g*\nu,\psi\rangle_{\omega}=\langle
g,\nu\cdot\psi\rangle_{\omega}=\langle\nu\cdot\psi,g\rangle_{{\cal
L}^{\infty}-{\cal L}^{1}}.$

Hence, $C_{0}(G,\omega)$ is left introverted and
$\langle\mu{\scriptscriptstyle\square}\nu,\psi\rangle=\langle\mu,\nu{\scriptscriptstyle\square}\psi\rangle=\langle\mu,\nu\cdot\psi\rangle=\langle\mu*\nu,\psi\rangle$,
where we have again used (14). Similarly, $C_{0}(G,\omega^{-1})$ is right
introverted and $\mu*\nu=\mu\diamond\nu$. ∎

Let ${\cal S}(\omega^{-1})$ be a left introverted subspace of ${\cal
L}^{\infty}(G,\omega^{-1})$ such that $C_{0}(G,\omega^{-1})\preceq{\cal
S}(\omega^{-1})\preceq LUC(G,\omega^{-1})$ and define

$\displaystyle\Theta:{\cal M}(G,\omega)\rightarrow{\cal S}(\omega^{-1})^{*}\ \
{\rm by}\ \ \langle\Theta(\nu),\psi\rangle_{{\cal S}^{*}-{\cal
S}}=(\nu\cdot\psi)(e_{G})=\int\psi\,\chi_{F_{\nu}}\,d\nu$ (15)

where $\nu\in{\cal M}(G,\omega)$, $\psi\in{\cal S}(\omega^{-1})$ and $F_{\nu}$
is any $\sigma$-compact set on which $\nu$ is concentrated. By Proposition
2.5, $\Theta$ is well-defined and
$|\langle\Theta(\nu),\psi\rangle|\leq\|\nu\cdot\psi\|_{\infty,\omega^{-1}}\leq\|\nu\|_{\omega}\|\psi\|_{\infty,\omega^{-1}}$,
so $\|\Theta(\nu)\|\leq\|\nu\|_{\omega}$; by equation (13),
$\Theta(\nu)\Large{|}_{C_{0}(G,\omega^{-1})}=\nu$, so
$\|\Theta(\nu)\|=\|\nu\|_{\omega}$. Thus, $\Theta$ is a linear isometry.

Let $so_{l}$ and $so_{r}$ denote the left and right strict topologies on
${\cal M}(G,\omega)$ taken with respect to the ideal ${\cal L}^{1}(G,\omega)$,
i.e., the locally convex topologies respectively generated by the semi-norms
$p_{g}(\nu)=\|g*\nu\|$ and $q_{g}(\nu)=\|\nu*g\|$ for $g\in{\cal
L}^{1}(G,\omega),\nu\in{\cal M}(G,\omega)$. Since ${\cal L}^{1}(G,\omega)$ has
a contractive approximate identity, (the unit ball of) ${\cal
L}^{1}(G,\omega)$ is $so_{l}/so_{r}$-dense in (the unit ball of) ${\cal
M}(G,\omega)$. Observe that when ${\cal S}(\omega^{-1})\preceq
LUC(G,\omega^{-1})$ is a ${\cal L}^{1}(G,\omega)$-submodule of ${\cal
L}^{\infty}(G,\omega^{-1})$, by Lemma 2.3(d) and the Cohen factorization
theorem [1, Theorem 11.10], ${\cal S}(\omega^{-1})={\cal
S}(\omega^{-1})\cdot{\cal L}^{1}(G,\omega)$. Also note that
$LUC(G,\omega^{-1})$ is always left introverted in ${\cal
L}^{\infty}(G,\omega^{-1})$ by Lemma 2.3 and [3, Proposition 5.9]. In the non-
weighted case and when $\omega\geq 1$, the final statement in Proposition 2.9,
which simplifies Arens product calculations, is [13, Lemma 3] and [3,
Proposition 7.21], respectively.

######  Proposition 2.9.

Suppose that ${\cal S}(\omega^{-1})$ is a left [right] introverted subspace of
${\cal L}^{\infty}(G,\omega^{-1})={\cal L}^{1}(G,\omega)^{*}$ and
$C_{0}(G,\omega^{-1})\preceq{\cal S}(\omega^{-1})\preceq LUC(G,\omega^{-1})\
[RUC(G,\omega^{-1})]$. Then $\Theta:{\cal M}(G,\omega)\hookrightarrow{\cal
S}(\omega^{-1})^{*}$ is a $so_{l}$-weak∗ [$so_{r}$-weak∗] continuous isometric
homomorphic embedding into $Z_{t}({\cal S}(\omega^{-1})^{*})$ that extends
$\eta_{\cal S}:{\cal L}^{1}(G,\omega)\rightarrow{\cal S}(\omega^{-1})^{*}$.
Moreover, $(n{\scriptscriptstyle\square}\psi)(s)=\langle n,\psi\cdot s\rangle$
for any $n\in{\cal S}(\omega^{-1})^{*}$, $\psi\in{\cal S}(\omega^{-1})$ and
$s\in G$; hence, ${\cal S}(\omega^{-1})$ is introverted as a subspace of
$\ell^{\infty}(G,\omega^{-1})=\ell^{1}(G,\omega)^{*}$, the Arens product on
${\cal S}(\omega^{-1})^{*}$ agrees under either interpretation, and $\Theta$
also extends $\eta_{\cal S}:\ell^{1}(G,\omega)\hookrightarrow{\cal
S}(\omega^{-1})^{*}$.

###### Proof.

If $g\in{\cal L}^{1}(G,\omega)={\cal L}^{1}(\omega\lambda)$, $g$ vanishes off
a $\sigma$-compact set $F_{g}$, and therefore $g=g\lambda\in{\cal
M}(G,\omega)$ is concentrated on $F_{g}$; hence, for $\psi\in{\cal
S}(\omega^{-1})$,

$\langle\Theta(g),\psi\rangle_{{\cal S}^{*}-{\cal
S}}=\int\psi\,\chi_{F_{g}}\,d(g\lambda)=\int\psi
g\,d\lambda=\langle\psi,g\rangle_{{\cal L}^{\infty}-{\cal
L}^{1}}=\langle\eta_{\cal S}(g),\psi\rangle_{{\cal S}^{*}-{\cal S}}.$

For $f\in{\cal L}^{1}(G,\omega)$, $\nu\in{\cal M}(G,\omega)$ and $\psi\in{\cal
S}(\omega^{-1})$,

$\displaystyle\langle\Theta(f){\scriptscriptstyle\square}\Theta(\nu),\psi\rangle_{{\cal
S}^{*}-{\cal S}}$ $\displaystyle=$ $\displaystyle\langle\eta_{\cal
S}(f),\Theta(\nu){\scriptscriptstyle\square}\psi\rangle_{{\cal S}^{*}-{\cal
S}}=\langle\Theta(\nu){\scriptscriptstyle\square}\psi,f\rangle_{{\cal
L}^{\infty}-{\cal L}^{1}}=\langle\Theta(\nu),\psi\cdot f\rangle_{{\cal
S}^{*}-{\cal S}}$ $\displaystyle=$ $\displaystyle\nu\cdot(\psi\cdot
f)(e_{G})=(\nu\cdot\psi)\cdot
f(e_{G})=\langle\nu\cdot\psi,f*\delta_{e_{G}}\rangle_{{\cal L}^{\infty}-{\cal
L}^{1}}$ $\displaystyle=$ $\displaystyle\langle\psi,f*\nu\rangle_{{\cal
L}^{\infty}-{\cal L}^{1}}=\langle\eta_{\cal S}(f*\nu),\psi\rangle_{{\cal
S}^{*}-{\cal S}}=\langle\Theta(f*\nu),\psi\rangle_{{\cal S}^{*}-{\cal S}},$

where we have used (12). Suppose that $\nu_{i}\rightarrow\nu$ $so_{l}$.
Writing $\psi\in{\cal S}(\omega^{-1})$ as $\psi=\phi\cdot g$ for some
$\phi\in{\cal S}(\omega^{-1})$ and $g\in{\cal L}^{1}(G,\omega)$,

$\displaystyle\langle\Theta(\nu_{i})-\Theta(\nu),\psi\rangle_{{\cal
S}^{*}-{\cal
S}}=\langle\Theta(g){\scriptscriptstyle\square}\Theta(\nu_{i}-\nu),\phi\rangle_{{\cal
S}^{*}-{\cal S}}=\langle\Theta(g*(\nu_{i}-\nu)),\phi\rangle_{{\cal
S}^{*}-{\cal S}}\rightarrow 0.$

Hence, $\Theta$ is $so_{l}$-weak∗ continuous. Let $\mu,\nu\in{\cal
M}(G,\omega)$ and let $(h_{i})$ be a net in ${\cal L}^{1}(G,\omega)$ such that
$so_{l}-\lim h_{i}=\mu$. Then $so_{l}-\lim h_{i}*\nu=\mu*\nu$, so

$\Theta(\mu){\scriptscriptstyle\square}\Theta(\nu)={\rm
wk}^{*}-\lim\Theta(h_{i}){\scriptscriptstyle\square}\Theta(\nu)={\rm
wk}^{*}-\lim\Theta(h_{i}*\nu)=\Theta(\mu*\nu).$

Identify the Banach algebra ${\cal M}(G,\omega)$ with its copy $\Theta({\cal
M}(G,\omega))$ in ${\cal S}(\omega^{-1})^{*}$. Since ${\cal
S}(\omega^{-1})={\cal S}(\omega^{-1})\cdot{\cal L}^{1}(G,\omega)$ is a right
${\cal M}(G,\omega)$-module, ${\cal S}(\omega^{-1})^{*}$ is a left dual ${\cal
M}(G,\omega)$-module, and the proof of [8, Lemma 1.4] shows that
$\mu{\scriptscriptstyle\square}n=\mu\cdot n$ for $\mu\in{\cal M}(G,\omega)$
and $n\in{\cal S}(\omega^{-1})^{*}$; hence, $\Theta$ maps into $Z_{t}({\cal
S}(\omega^{-1})^{*})$. For $n\in{\cal S}(\omega^{-1})^{*}$, $\psi\in{\cal
S}(\omega^{-1})$ and $s\in G$,
$(n{\scriptscriptstyle\square}\psi)(s)=\langle\delta_{s},n{\scriptscriptstyle\square}\psi\rangle=\langle\delta_{s}{\scriptscriptstyle\square}n,\psi\rangle=\langle\delta_{s}\cdot
n,\psi\rangle=\langle n,\psi\cdot\delta_{s}\rangle=\langle n,\psi\cdot
s\rangle.$ The final line is now easily verified. ∎

For a Banach algebra $A$, the space $WAP(A^{*})$ of weakly almost periodic
functionals on $A$ is a left and right introverted subspace of $A^{*}$ such
that for every $m,n\in WAP(A^{*})^{*}$,
$m{\scriptscriptstyle\square}n=m\diamond n$ [3, Proposition 3.11]. Thus,
$WAP(A^{*})^{*}$ is a dual Banach algebra. Moreover, $WAP(A^{*})^{*}$
satisfies the following universal property [16, Theorem 4.10].

######  Theorem 2.10.

(Runde) If ${\mathfrak{B}}$ is a dual Banach algebra and
$\varphi:A\rightarrow{\mathfrak{B}}$ is a continuous algebra homomorphism,
then there is a unique weak∗-weak∗ continuous algebra homomorphism
$\varphi_{WAP}:WAP(A^{*})^{*}\rightarrow{\mathfrak{B}}$ such that
$\varphi_{WAP}\circ\eta_{WAP}=\varphi$.

Taking $A_{\omega}={\cal L}^{1}(G,\omega)$, it follows that the embedding
${\rm id}:{\cal L}^{1}(G,\omega)\hookrightarrow{\cal M}(G,\omega)$ determines
a unique weak∗-weak∗ continuous homomorphism
$P:WAP(A_{\omega}^{*})^{*}\rightarrow{\cal M}(G,\omega)$ such that
$P\circ\eta_{WAP}={\rm id}$. Letting $P_{*}:C_{0}(G,\omega^{-1})\rightarrow
WAP(A_{\omega}^{*})\preceq{\cal L}^{\infty}(G,\omega^{-1})$ denote the predual
mapping of $P$, $\langle P_{*}\psi,g\rangle_{{\cal L}^{\infty}-{\cal
L}^{1}}=\langle P\circ\eta_{WAP}(g),\psi\rangle=\langle\psi,g\rangle_{{\cal
L}^{\infty}-{\cal L}^{1}}$ for $\psi\in C_{0}(G,\omega^{-1})$, $g\in{\cal
L}^{1}(G,\omega)$. Hence, $C_{0}(G,\omega^{-1})\preceq WAP(A_{\omega}^{*})$.
Moreover, by [3, Proposition 3.12] and Lemma 2.3,
$WAP(A_{\omega}^{*})\preceq(LUC\cap RUC)(G,\omega^{-1})$. Hence, we have the
following immediate corollary to Proposition 2.9.

######  Corollary 2.11.

The map $\Theta:{\cal M}(G,\omega)\hookrightarrow WAP(A_{\omega}^{*})^{*}$, as
defined in (15), is a $so_{l}$-weak∗ and $so_{r}$-weak∗ continuous isometric
homomorphic embedding that extends $\eta_{WAP}:{\cal
L}^{1}(G,\omega)\hookrightarrow WAP(A_{\omega}^{*})^{*}$.

As shown in [3], $WAP(A_{\omega}^{*})$ may fail to equal
$WAP(G,\omega^{-1})=\left\\{f:{f\over\omega}\in WAP(G)\right\\}$. Our final
two results are needed in [12]. Corollary 2.12 improves [10, Theorem 5.6] in
the case of ${\cal L}^{1}(G,\omega)$:

######  Corollary 2.12.

Let ${\mathfrak{B}}$ be a dual Banach algebra, $\varphi:{\cal
L}^{1}(G,\omega)\rightarrow{\mathfrak{B}}$ a bounded homomorphism. Then there
is a unique $so_{l}$-weak∗ and $so_{r}$-weak∗ continuous homomorphic extension
$\widetilde{\varphi}:{\cal M}(G,\omega)\rightarrow{\mathfrak{B}}$ of
$\varphi$.

###### Proof.

Letting $\varphi_{WAP}:WAP(A_{\omega}^{*})^{*}\rightarrow{\mathfrak{B}}$ be
the weak∗-weak∗ continuous extension of $\varphi$ from Theorem 2.10 and
$\Theta:{\cal M}(G,\omega)\hookrightarrow WAP(A_{\omega}^{*})^{*}$ the
$so_{l}/so_{r}$-weak∗ continuous embedding from Corollary 2.11,
$\widetilde{\varphi}:=\varphi_{WAP}\circ\Theta$ is the desired extension;
uniqueness follows from the $so_{l}$-density of ${\cal L}^{1}(G,\omega)$ in
${\cal M}(G,\omega)$. ∎

######  Corollary 2.13.

Let ${\mathfrak{B}}$ be a dual Banach algebra, $\varphi:{\cal
M}(G,\omega)\rightarrow{\mathfrak{B}}$ a bounded homomorphism that is
$so_{l}$-weak∗ continuous on the unit ball of ${\cal M}(G,\omega)$. Then
$\varphi$ is $so_{l}$-weak∗ and $so_{r}$-weak∗ continuous on all of ${\cal
M}(G,\omega)$.

###### Proof.

By Corollary 2.12, the restriction, $\varphi_{1}$, of $\varphi$ to ${\cal
L}^{1}(G,\omega)$ has a $so_{l}$/$so_{r}$-weak∗ continuous extension
$\widetilde{\varphi_{1}}:{\cal M}(G,\omega)\rightarrow{\mathfrak{B}}$. As
noted before, ${\cal L}^{1}(G,\omega)_{\|\cdot\|\leq 1}$ is $so_{l}$-dense in
${\cal M}(G,\omega)_{\|\cdot\|\leq 1}$, so $\varphi=\widetilde{\varphi_{1}}$
on ${\cal M}(G,\omega)_{\|\cdot\|\leq 1}$ and therefore on ${\cal
M}(G,\omega)$. ∎

######  Remark 2.14.

Suppose that $(H,\omega_{H})$ is another weighted locally compact group and
$\varphi:{\cal M}(G,\omega)\rightarrow{\cal M}(H,\omega_{H})$ is a bounded
algebra isomorphism. By [8, Lemma 3.3] — which applies, as written, to ${\cal
M}(G,\omega)$ — $\varphi$ is $so_{l}$-weak∗ continuous on bounded subsets of
${\cal M}(G,\omega)$. By Corollary 2.13, $\varphi$ is $so_{l}/so_{r}$-weak∗
continuous on all of ${\cal M}(G,\omega)$.

Acknowledgements: The author is grateful to Fereidoun Ghahramani for helpful
discussions regarding the topic of this paper.

## References

  * [1] F. Bonsall, and J. Duncan, Complete normed algebras, Springer-Verlag, 1973.
  * [2] H.G. Dales, Banach algebras and automatic continuity, London Math. Soc. Monographs, Volume 24, Clarendon Press, Oxford, 2000.
  * [3] H.G. Dales and A. T.-M. Lau, The second duals of Beurling algebras, Mem. Amer. Math. Soc. 177 (2005), no. 836.
  * [4] J.M.G. Fell and R.S. Doran, Representations of $*$-algebras, locally compact groups, and Banach $*$-algebraic bundles Vol. 1, Pure and Applied Mathematics, 125. Academic Press, Inc., 1988.
  * [5] F. Ghahramani, Weighted group algebra as an ideal in its second dual space, Proc. Amer. Math. Soc. 90 (1984), no. 1, 71-76.
  * [6] E. Hewitt and K.A. Ross, Abstract harmonic analysis, Springer-Verlag, New York, 1963.
  * [7] F. Ghahramani, Compact elements of weighted group algebras, Pacific J. Math. 113 (1984), no. 1, 77-84.
  * [8] F. Ghahramani and S. Zadeh, Bipositive isomorphisms between Beurling algebras and between their second dual algebras, Canad. J. Math. 69 (2017), no. 1, 3-20
  * [9] N. Gr${\rm\o{}}$nb$\ae$k, Amenability of weighted convolution algebras on locally compact groups. Trans. Amer. Math. Soc. 319(1990), no. 2, 765–775.
  * [10] M. Ilie and R. Stokke, Weak∗-continuous homomorphisms of Fourier-Stieltjes algebras, Math. Proc. Cambridge Philos. Soc., 145 (2008), 107-120.
  * [11] E. Kaniuth, A course in commutative Banach algebras, Springer, New York, 2009.
  * [12] M. Kroeker, A. Stephens, R. Stokke and R. Yee, Positive homomorphisms of Beurling algebras, in preparation.
  * [13] A. T.-M. Lau, Operators which commute with convolutions on sub- spaces of $L^{\infty}(G)$, Colloquium Math., 39 (1978), 351–359.
  * [14] T.W. Palmer, Banach algebras and the general theory of $*$-algebras. Vol. I., Cambridge University Press, Cambridge, 1994.
  * [15] H. Reiter and J.D. Stegeman, Classical harmonic analysis and locally compact groups, second ed., London Math. Soc. Monographs, Volume 22, Clarendon Press, Oxford, 2000.
  * [16] V. Runde, Dual Banach algebras: Connes-amenability, normal, virtual diagonals, and injectivity of the predual bimodule, Math. Scand.  95 (2004), no. 1, 124–144.
  * [17] E. Samei, Weak amenability and 2-weak amenability of Beurling algebras, J. Math. Anal. Appl. 346 (2008), no. 2, 451-467.
  * [18] V. Shepelska and Y. Zhang, Non-weakly amenable Beurling algebras, Indiana Univ. Math. J. 67 (2018), no. 1, 119-150.
  * [19] S. Zadeh, Isomorphisms of Banach algebras associated with locally compact groups, PhD thesis, University of Manitoba, Canada, 2015.
  * [20] S. Zadeh, Isometric isomorphisms of Beurling algebras, J. Math. Anal. Appl. 438 (2016), 1-13.

Department of Mathematics and Statistics, University of Winnipeg, 515 Portage
Avenue, Winnipeg, MB, R3B 2E9, Canada

email<EMAIL_ADDRESS>