Introduction
Let
G be a locally compact group with a fixed left Haar measure λ and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
be a weight function on
G; that is a Borel measurable function
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image004.gif)
with
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image006.gif)
for all
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image008.gif)
. We denote by
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
the set of all measurable functions
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image012.gif)
such that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image014.gif)
; the group algebra of
G as defined in [2]. Then
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
with the convolution product “*” and the norm
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif)
defined by
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image018.gif)
is a Banach algebra known as Beurling algebra. We denote by
n(
G,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
) the topology generated by the norm
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image016.gif)
. Also, let
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
denote the space of all measurable functions 𝑓 with
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image022.gif)
, the Lebesgue space as defined in [2].
Then
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
with the product
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image024.gif)
defined by
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image026.gif)
, the norm
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif)
defined by
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image030.gif)
, and the complex conjugation as involution is a commutative
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image032.gif)
algebra. Moreover,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
is the dual of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
. In fact, the mapping
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image034.gif)
is an isometric isomorphism.
We denote by
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image036.gif)
the
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image038.gif)
-subalgebra of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
consisting of all functions 𝘨 on
G such that for each
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image040.gif)
, there is a compact subset
K of
G for which
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image042.gif)
. For a study of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image044.gif)
in the unweighted case see [3,6].
We introduce and study a locally convex topology
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif)
on
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
such that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif)
can be identified with the strong dual of
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
. Our work generalizes some interesting results of [15] for group algebras to a more general setting of weighted group algebras. We also show that (
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif)
) could be a normable or bornological space only if
G is compact. Finally, we prove that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif)
is complemented in
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image020.gif)
if and only if G is compact. For some similar recent studies see [4,7,8,10,12-14]. One may be interested to see the work [9] for an application of these results.
Main results
We denote by
𝒞 the set of increasing sequences of compact subsets of G and by
ℛ the set of increasing sequences
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image052.gif)
of real numbers in
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image054.gif)
divergent to infinity. For any
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image056.gif)
and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image058.gif)
, set
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image060.gif)
and note that
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif)
is a convex balanced absorbing set in the space
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
. It is easy to see that the family
𝒰 of all sets
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image062.gif)
is a base of neighbourhoods of zero for a locally convex topology on
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image064.gif)
see for example [16]. We denote this topology by
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif)
. Here we use some ideas from [15], where this topology has been introduced and studied for group algebras.
Proposition 2.1 Let
G be a locally compact group, and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif)
be a weight function on
G. The norm topology
n(
G,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif)
) on
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
coincides with the topology
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image046.gif)
if and only if
G is compact.
Proposition 2.2 Let
G be a locally compact group, and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif)
be a weight function on
G. Then the dual of (
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif)
) endowed with the strong topology can be identified with
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image048.gif)
endowed with
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image028.gif)
-topology.
Proposition 2.3 Let
G be a locally compact group, and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif)
be a weight function on
G. Then the following assertions are equivalent:
a) (
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif)
) is barrelled.
b) (
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif)
) is bornological.
c) (
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image010.gif)
,
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image050.gif)
) is metrizable.
d)
G is compact.
Proposition 2.4 Let
G be a locally compact group, and
![](file:///C:/Users/1/AppData/Local/Temp/msohtmlclip1/01/clip_image066.gif)
be a weight function on
G. Then is not complemented in .
./files/site1/files/52/10.pdf
Type of Study:
Original Manuscript |
Subject:
alg Received: 2016/10/25 | Revised: 2019/12/16 | Accepted: 2018/06/27 | Published: 2019/11/18 | ePublished: 2019/11/18