Derivations on the algebra of Rajchman measures

Abstract

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.