application of positive linear functionl The space $L^{1}(\mathbb R)$ is a commutative Banach algebra under convolution. A linear functional $F$ on $L^{1}(\mathbb R)$ is positive if $F(f^*f)\geq 0$ for $f\in L^{1}(\mathbb R).$ (where $f^*(x)=\bar{f(x)}$).  (Actually, $L^{1}$ is a Banach star algebra with the involution $f^*(x)=\overline{f(x)}$).  

My naive Questions are:
  (1)  What are examples of positive linear functional on $L^{1}(\mathbb R)$?

(2) Does there exists positive linear functional on $L^{1}(\mathbb R)$ which is not continuous? If answer is no, then can find some examples of Banach star algebra where positive linear functional are discontinuous?
(3) What is an importance(and motivation) of the study of positive linear functional on $L^{1}(\mathbb R);$ can you illustrate some application(here I mean in some branch of mathematics only) of it?
 A: I can't bring myself to write $fg$ for the convolution of $f$ and $g$. So I'm going to write $f\mapsto f'$ for the involution, so I can write $f*g$ for the convolution.
Are you certain you got the definition of $f'$ straight? What would make much more sense to me would be $$f'(t)=\overline{f(-t).}$$
That seems to me is the "standard" involution on $L^1$. Standard because it does good things, for example if $\hat f$ denotes the Fourier transform then $$\widehat{f'}=\overline{\hat f}.$$Hence $$\widehat{f'*f}=|\hat f|^2,$$and hence there are lots of positive functionals: If $\mu\ge0$ is a finite Borel measure on $\mathbb R$ and we define $$T_\mu f=\int\hat f\,d\mu$$then $T_\mu$ is positive.
It's easy to show that each $T_\mu$ is continuous. I tend to suspect that every positive linear functional on $L^1$ is $T_\mu$ for some $\mu$ as above, although I don't have details of the proof (seems like we need some sort of Hahn-Banach extension for positive functionals).
I can say that one obvious way to construct a discontinuous positive functional doesn't work: We might try $T_\mu$ where $\mu\ge0$ but $\mu(\mathbb R)=\infty$. That doesn't work; one can show that there must exist $f$ such that $T_\mu(f'*f)=\infty$.
