  In this paper, we examine Atiyah's Hermitian axiom for two-dimensional
complex topological quantum field theories. Building on the correspondence
between 2D TQFTs and Frobenius algebras, we find the algebraic objects
corresponding to Hermitian and unitary TQFTs respectively and prove structure
theorems about them. We then clarify a few older results on unitary TQFTs using
our structure theorems.
