Amenability properties of Rajchman algebras Mahya Ghandehari 43A3046L0747B4746J10Rajchman algebraFourier algebraFourier-Stieltjes algebralocally compact groupsamenabilityoperator amenabilitybounded approximate identity Rajchman measures of locally compact abelian groups have been studied for almost a century now, and they play an important role in the study of trigonometric series. Eymard's influential work allowed generalizing these measures to the case of \emph{non-abelian} locally compact groups $G$. The Rajchman algebra of $G$, which we denote by $B_0(G)$, is the set of all elements of the Fourier--Stieltjes algebra that vanish at infinity. In the present article, we characterize the locally compact groups that have amenable Rajchman algebras. We show that $B_0(G)$ is amenable if and only if $G$ is compact and almost abelian. On the other extreme, we present many examples of locally compact groups, such as non-compact abelian groups and infinite solvable groups, for which $B_0(G)$ fails to even have an approximate identity. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4622 10.1512/iumj.2012.61.4622 en Indiana Univ. Math. J. 61 (2012) 1369 - 1392 state-of-the-art mathematics http://iumj.org/access/