Growth and smooth spectral synthesis in the Fourier algebras of Lie groups. Jean Ludwig Department of Mathematics, University of Metz, F-57045 Metz, France Lyudmila Turowska ∗ Department of Mathematics, Chalmers University of Technology, SE-412 96 G¨oteborg, Sweden Abstract Let G be a Lie group and A(G) the Fourier algebra of G. In this paper we decribe sufficient conditions for complex-valued functions to operate on elements u ∈ A(G) satisfying certain degrees of differentiability in terms of the dimension of the group G. Furthermore, generalizing a result of Kirsch and M¨ uller [Ark. Mat. 18 (1980), no. 2, 145–155] we prove that closed subsets E of a smooth m-dimensional sub-manifold of a Lie group G having a certain cone property are sets of smooth spectral synthesis. For such sets we give an estimate of the degree of nilpotency of the quotient algebra I A (E)/J A (E), where I A (E) and J A (E) are the largest and the smallest closed ideal in A(G) with hull E. 1 Introduction In this paper we study two questions concerning the Fourier algebra A(G) of a (non- commutative) Lie group G. The first one deals with the functional calculus in A(G) and the other one with problems of spectral synthesis in the same algebra. Functional calculus is one of the basic tools in the theory of Banach algebras and its applications. In particular, it plays a fundamental role in some parts of harmonic * Corresponding author. E-mail address: turowska@math.chalmers.se. Fax: +46 31 161973 0 2000 Mathematics Subject Classification: 46J10 (Primary), 46E15, 43A45, 22E30 (Secondary) 1