  In his Inventiones paper, Ziller (Invent. Math: 1-22, 1977) computed the
integral homology as a graded abelian group of the free loop space of compact,
globally symmetric spaces of rank 1. Chas and Sullivan (String Topology,
1999)showed that the homology of the free loop space of a compact closed
orientable manifold can be equipped with a loop product and a BV-operator
making it a Batalin-Vilkovisky algebra. Cohen, Jones and Yan (The loop homology
algebra of spheres and projective spaces, 2004) developed a spectral sequence
which converges to the loop homology as a spectral sequence of algebras. They
computed the algebra structure of the loop homology of spheres and complex
projective spaces by using Ziller's results and the method of Brown-Shih (Ann.
of Math. 69:223-246, 1959, Publ. Math. Inst. Hautes \'Etudes Sci. 3: 93-176,
1962). In this note we compute the loop homology algebra by using only spectral
sequences and the technique of universal examples. We therefore not only obtain
Zillers' and Brown-Shihs' results in an elementary way, we also replace the
roundabout computations of Cohen, Jones and Yan (The loop homology algebra of
spheres and projective spaces, 2004) making them independent of Ziller's and
Brown-Shihs' work. Moreover we offer an elementary technique which we expect
can easily be generalized and applied to a wider family of spaces, not only the
globally symmetric ones.
