A locally Convex Topology on the Beurling Algebras

Maghsoudi, Saeid

doi:10.29252/mmr.5.2.221

Volume 5, Issue 2 (Vol. 5,No. 2 2019) mmr 2019, 5(2): 221-228 | Back to browse issues page

‎ 10.29252/mmr.5.2.221

‎ 20.1001.1.25882546.1398.5.2.1.5

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Maghsoudi S. A locally Convex Topology on the Beurling Algebras. mmr 2019; 5 (2) :221-228
URL: http://mmr.khu.ac.ir/article-1-2556-en.html

A locally Convex Topology on the Beurling Algebras

Saeid Maghsoudi ^*

University of Zanjan , s_maghsodi@znu.ac.is

Abstract: (4100 Views)

Introduction
Let G be a locally compact group with a fixed left Haar measure λ and

be a weight function on G; that is a Borel measurable function

with

for all

. We denote by

the set of all measurable functions

such that

; the group algebra of G as defined in [2]. Then

with the convolution product “*” and the norm

defined by

is a Banach algebra known as Beurling algebra. We denote by n(G,

) the topology generated by the norm

. Also, let

denote the space of all measurable functions 𝑓 with

, the Lebesgue space as defined in [2].
Then

with the product

defined by

, the norm

defined by

, and the complex conjugation as involution is a commutative

algebra. Moreover,

is the dual of

. In fact, the mapping

is an isometric isomorphism.
We denote by

the

-subalgebra of

consisting of all functions 𝘨 on G such that for each

, there is a compact subset K of G for which

. For a study of

in the unweighted case see [3,6].
We introduce and study a locally convex topology

such that

can be identified with the strong dual of

. Our work generalizes some interesting results of [15] for group algebras to a more general setting of weighted group algebras. We also show that (

) could be a normable or bornological space only if G is compact. Finally, we prove that

is complemented in

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

of real numbers in

divergent to infinity. For any

and

, set

and note that

is a convex balanced absorbing set in the space

. It is easy to see that the family 𝒰 of all sets

is a base of neighbourhoods of zero for a locally convex topology on

see for example [16]. We denote this topology by

. 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