how big is group C*-algebra of $\mathbb{R}^n$?

Question

We know that the group C*-algebra of the group $\mathbb{R}^n$, $C^*(\mathbb{R}^n)$, is the completion of $L^1(\mathbb{R}^n)$ inside $B(L^2(\mathbb{R}^n))$. My question is, does $C^*(\mathbb{R}^n)$ contain $L^p(\mathbb{R}^n)$, for $1\le p \le 2$?

Answer 1

No, it doesn't. In fact, $C^{\ast}(\mathbb{R}^n)$ can be identified with $C_{0}(\mathbb{R}^{n})$ via the Fourier transform, which also maps $L^{2}$ onto itself $L^{2}$. There are many examples of $L^2$-functions that do not belong to $C_{0}$.

A more down to earth approach would be the following: if a function $f$ belongs to the group $C^{\ast}$-algebra then $f \ast g \in L^2(\mathbb{R}^n)$ for any $g \in L^{2}(\mathbb{R}^n)$. This property, again via Fourier transform, is equivalent to the fact that $\widehat{f}\cdot g \in L^{2}(\mathbb{R}^n)$ for any $g\in L^{2}(\mathbb{R}^n)$. This means at least that $\widehat{f}$ is bounded, which is not the case for many functions in $L^{p}$ for $p>1$.