1. Generally in representation theory it is in practice to consider the unitary representations of a locally compact group. For example look into the book on “ABSTRACT HARMONIC ANALYSIS” by “G.B.FOLLAND”. One of the milestones of representation theory is Schur’s lemma. This Schur’s lemma when applied to a locally compact abelian group results in the fact that all irreducible unitary representations are one dimensional. As the unitary operators on the one dimensional space of complex numbers is the circle, we consider only the group homomorphisms from the LCA group G to the circle group. This set is on its own an Abelian group under pointwise multiplication and in fact a locally compact abelian group with the topology of uniform convergence on compact sets. This group is called as the DUAL GROUP genrally denoted as G^. On the other hand, consider the Banach algebra L1(G) for a locally compact group with respect to the Haar measure, pointwise addition and convolution as product. Notice that L1(G) is commutative if and only if G is Abelian. Thus for a locally compact abelian group L1(G) is a commutative Banach algebra and therefore it makes sense to speak about the spectrum of the algebra which the set of all multiplicative linear functionals on the Banach algebra L1(G). It turns that the spectrum of L1(G) is G^. Under these natural identifications and the natural topology on it, I suppose the question regarding the divergence makes sense.
2. Well as for as the definition of a L-subalgebra one has to notice the following fact that M(G) is a L\infty(G) module. I think that this much is sufficient to explain the definition.

From: fatima22_m <fatima22_m@...>
Subject: [harmonic] question
To: harmonicanalysis@yahoogroups.com
Date: Wednesday, 1 October, 2008, 8:03 PM

Dear members
Thanks for your reply to my questions
Here there are some questions
Let G is a locally compact abelian group and à is duel
group
In book Fourier analysis on groups that is written by Walter
Rudin it is said
That à is the set of all continuous homomorphism on G with the
absolute value one
But I have seen in some text that we suppose there is a net of
characters in which converge infinity I cannot understand when
this characters have absolute value one how we talk about
converging to infinity ?
Second question is about Ã(µ) what is the exact
definition of Ã(µ)?
What is the relation between Ã(µ) and L^1 (ì) and L^1 (ì)^?
In one of the paper that I am studying introduce L-
subalgebra in this way
"a norm closed subspace of measures which is closed under
the operation Of multiplication by bounded
measurable functions please explain
About the meaning of definition
Does Ã(µ) separate the points of L^1 (ì)?
Why (C(Ã(µ) )=) &#773;L^1 (ì)^