  The lattice definition of the two-dimensional topological quantum field
theory [Fukuma, {\em et al}, Commun.~Math.~Phys.\ {\bf 161}, 157 (1994)] is
generalized to arbitrary (not necessarily orientable) compact surfaces. It is
shown that there is a one-to-one correspondence between real associative
$*$-algebras and the topological state sum invariants defined on such surfaces.
The partition and $n$-point functions on all two-dimensional surfaces
(connected sums of the Klein bottle or projective plane and $g$-tori) are
defined and computed for arbitrary $*$-algebras in general, and for the the
group ring $A=\R[G]$ of discrete groups $G$, in particular.
