A locally Convex Topology on the Beurling Algebras

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   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  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 We denote by   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 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