We develop the theory of semisimple weak Hopf algebras and obtain analogues
of a number of classical results for ordinary semisimple Hopf algebras. We
prove a criterion for semisimplicity and analyze the square of the antipode S^2
of a semisimple weak Hopf algebra A. We explain how the Frobenius-Perron
dimensions of irreducible A-modules and eigenvalues of S^2 can be computed
using the inclusion matrix associated to A. A trace formula of Larson and
Radford is extended to a relation between the global and Frobenius-Perron
dimensions of A. Finally, an analogue of the Class Equation of Kac and Zhu is
established and properties of $A$-module algebras and their dimensions are
studied.
