Let \(G\) be a locally compact group. Denote by \(C^*(G)\) the full \(C^*\)-algebra of \(G\), and by \(\mathcal K\) the algebra of compact operators on the infinite-dimensional separable Hilbert space. The main result of the paper runs as follows: let \(G\) be a semisimple Lie group with finite centre; there is no nonzero projection in \(C^*(G)\otimes{\mathcal K}\) if and only if the adjoint group of \(G\) has at least one simple factor isomorphic to an odd-dimensional Lorentz group \(SO_ 0(2n+1,1)\). If \(G\) is almost simple, this is still equivalent to the vanishing of \(K_ 0(C^*(G))\). These results are proved after a preliminary study of minimal projections in \(L^ 1(G)\) and in \(C^*(G)\) (in connection with integrable representations and with Kazhdan's property (T)).