  We describe the representation theory of C*-crossed-products of a unital
C*-algebra A by the cyclic group of order 2. We prove that there are two main
types of irreducible representations for the crossed-product: those whose
restriction to A is irreducible and those who are the sum of two unitarily
unequivalent representations of A. We characterize each class in term of the
restriction of the representations to the fixed point C*-subalgebra of A. We
apply our results to compute the K-theory of several crossed-products of the
free group on two generators.
