On idempotents of completely bounded multipliers of the Fourier algebra $A(G)$ Ana-Maria Stan 43A2243A4646L0746L1047L10Fourier algebraoperator spacecompletely bounded multiplierslacunary set Let $A(G)$ be the Fourier algebra of a locally compact group $G$ and $M_{cb}A(G)$ be the space of completely bounded multipliers of $A(G)$. We give a description of idempotents of $M_{cb}A(G)$ of norm one and then present a necessary condition for an idempotent to be in $M_{cb}A(G)$. In the last part of the paper, we discuss a class of free sets, called L-sets, whose characteristic function is an idempotent of $M_{cb}A(G)$, but not of $B(G)$. Indiana University Mathematics Journal 2009 text pdf 10.1512/iumj.2009.58.3452 10.1512/iumj.2009.58.3452 en Indiana Univ. Math. J. 58 (2009) 523 - 536 state-of-the-art mathematics http://iumj.org/access/