  For a locally compact Abelian group $G$, the algebra of Rajchman measures,
denoted by $M_0(G)$, is the set of all bounded regular Borel measures on $G$
whose Fourier transform vanish at infinity. In this paper, we investigate the
spectral structure of the algebra of Rajchman measures, and illustrate aspects
of the residual analytic structure of its maximal ideal space. In particular,
we show that $M_0(G)$ has a nonzero continuous point derivation, whenever $G$
is a non-discrete locally compact Abelian group. We then give the definition of
the Rajchman algebra for a general (not necessarily Abelian) locally compact
group, and prove that for a non-compact connected SIN group, the Rajchman
algebra admits a nonzero continuous point derivation. Moreover, we discuss the
analytic behavior of the spectrum of $M_0(G)$. Namely, we show that for every
non-discrete metrizable locally compact Abelian group $G$, the maximal ideal
space of $M_0(G)$ contains analytic discs.
