I have been working on a problem I found in a book. It starts by giving: $$A = \mathbb{T} = \{e^{i\theta} : \theta \in \mathbb{R} \}$$ along with the Haar measure $d\theta /2\pi$. From here it asks to find the norm on $C_0 (\mathbb{T}) \cap L^1 (\mathbb{T})$ and the space $C_0^* \mathbb{T}$.
I am fairly sure the norm is given by $max\{\mid f \mid _\mathbb{T} , \mid \mid L(f) \mid \mid _{C^* \mathbb{T}}\}$ (for $\mid f \mid _\mathbb{T}$ the usual supremum norm with respect to $\mathbb{T}$, $L(f)$ the convolution operator and $C^* \mathbb{T}$ the closure of $\{(f,\hat{f}) : f \in C_0 (\mathbb{T}) \cap L^1 (\mathbb{T})\}$) due to some information I found in the same book, however I am not entirly sure what the closure of $\{(f,\hat{f}) : f \in C_0 (\mathbb{T}) \cap L^1 (\mathbb{T})\}$ would look like and I was also wondering if the norm could be simplified?
There is also a question following asking to find the Plancherel measure on $\hat{\mathbb{T}}$. I am not sure how to go about this and any help would be appreciated.
Thanks in advance for any help.