On the quantum homology algebra of toric Fano manifolds

We study certain algebraic properties of the small quantum homology algebra for the class of symplectic toric Fano manifolds. In particular, we examine the semisimplicity of this algebra, and the more general property of containing a field as a direct summand. Our main result provides an easily verifiable sufficient condition for these properties which is independent of the symplectic form. Moreover, we answer two questions of Entov and Polterovich negatively by providing examples of toric Fano manifolds with non-semisimple quantum homology, and others in which the Calabi quasi-morphism is not unique.