Let $\mu$ be a Borel measure with finite variation on a locally compact abelian group $G$, let $\Gamma$ denote the dual group of $G$, and let $\hat \mu: \Gamma \to \mathbb{C}$ be the Fourier-Stieltjes transform of $\mu$. The measure $\mu$ induces a convolution operator $T_\mu$ on the space $L_1(G)$ defined by $T_\mu f = \mu \star f$, where $G$ is endowed with its Haar measure $m$.

The set $\hat\mu(\Gamma)$ is contained in the spectrum of $T$, and $T_\mu$ is said to have \emph{natural spectrum} when $\sigma(T_\mu)$ coincides with the closure of $\hat\mu(\Gamma)$. It is well-known that $T_\mu$ has natural spectrum when $\mu$ is discrete (its support is a countable union of atoms for $m$). See the Comment in page 619 of [M. Zafran. On the spectra of multipliers. Pacific J. Math. 47 (1973), 609-626].

I would like to have a direct proof of the last result, or a reference where I can find this proof.