arXiv:1901.08348v1 [math.GR] 24 Jan 2019 THE C ∗ -ALGEBRA OF THE CARTAN MOTION GROUPS REGEIBA HEDI AND RAHALI AYMEN Abstract. Let G0 = K ⋉ p be the Cartan motion groups. Under some assumption on G0, we describe the C ∗ -algebra C ∗ (G0) of G0 in terms of operator fields. 1. Introduction Let G be a locally compact group. We denote by  G the unitary dual of G. It well-known that  G equipped with the Fell topology (see [18, 19]). The first representation-theoretic question concerning the group G is the full parametrezation and topological identification of the dual  G. The C ∗ - algebra C ∗ (G) is the completion of the convolution algebra L 1 (G) equipped with the C ∗ -norm ‖.‖ C ∗ (G) , given by ‖f ‖ C ∗ (G) := sup π∈  G ‖π(f )‖ op . We denote by  C ∗ (G) the unitary dual of the C ∗ -algebra of G. Then we have the following bijection  C ∗ (G) ≃  G. Furthermore, the C ∗ -algebra C ∗ (G) of G can be identified with a subalgebra of the large C ∗ -algebra ℓ ∞ (  G) of bounded operator fields given by ℓ ∞ (  G) :=  F :  G −→  π∈  G B(Hπ ),π −→ F (π) ∈B(Hπ); ‖F ‖∞ := sup π∈  G ‖F (π)‖op < ∞  under the Fourier transform F defined on C ∗ (G) as follows: F (f )(π)= π(f ),π ∈  G, f ∈ C ∗ (G). Using the fact that F is an injective homomorphism of C ∗ (G) into ℓ ∞ (  G), then the C ∗ -algebra C ∗ (G) is isomorphic to a subalgebra D := F (C ∗ (G)) of elements in ℓ ∞ (  G) verifying some conditions. The elements of D must naturally fulfil is that of continuity. Then the parametrization and the descreption of the topology of  G are required to describe the C ∗ -algebra C ∗ (G) of G. In this context, we have some works in the literature , for example, J. Ludwig and L. Turowska have described in [16] the C ∗ -algebra of the Heisenberg group and of the thread-like Lie groups in terms of an algebra of operator fields defined over their dual spaces. The descreption of the C ∗ -algebra of the Euclidean motion group M n := SO(n) ⋉R n ,n ∈ N ∗ was established 1