A locally Convex Topology on the Beurling Algebras

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 on 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 be a weight function on G. The norm topology n(G,) on coincides with the topology if and only if G is compact.

Proposition 2.2 Let G be a locally compact group, and be a weight function on G. Then the dual of (,) endowed with the strong topology can be identified with endowed with -topology.

Proposition 2.3 Let G be a locally compact group, and be a weight function on G. Then the following assertions are equivalent:

a) (,) is barrelled.

b) (,) is bornological.

c) (,) is metrizable.

d) G is compact.

Proposition 2.4 Let G be a locally compact group, and be a weight function on G. Then is not complemented in ../files/site1/files/52/10.pdf