  We revisit Fourier's approach to solve the heat equation on the circle in the
context of (twisted) reduced group C*-algebras, convergence of Fourier series
and semigroups associated to negative definite functions. We introduce some
heat properties for countably infinite groups and investigate when they are
satisfied. Kazhdan's property (T) is an obstruction to the weakest property,
and our findings leave open the possibility that this might be the only one. On
the other hand, many groups with the Haagerup property satisfy the strongest
version. We show that this heat property implies that the associated heat
problem has a unique solution regardless of the choice of the initial datum.
