  We define a trivolution on a complex algebra $A$ as a non-zero
conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized
inverse of itself, that is, $\tau^3=\tau$. We give several characterizations of
trivolutions and show with examples that they appear naturally on many Banach
algebras, particularly those arising from group algebras. We give several
results on the existence or non-existence of involutions on the dual of a
topologically introverted space. We investigate conditions under which the dual
of a topologically introverted space admits trivolutions.
