  Partial actions of discrete groups on $C^*$-algebras and the associated
crossed products have been studied by Exel and McClanahan. We characterise
these crossed products in terms of the spectral subspaces of the dual coaction,
generalising and simplifying a theorem of Exel for single partial
automorphisms. We then use this characterisation to identify the Cuntz algebras
and the Toeplitz algebras of Nica as crossed products by partial actions.
