We show that the left regular representation of
a countably infinite (discrete) group admits no
finitedimensional invariant subspaces. We also
discuss a consequence of this fact, and the
reason for our interest in this statement. We
then formally state, as a `conjecture', a
possible generalisation of the above statement
to the context of fusion algebras. We prove the
validity of this conjecture in the case of the
fusion algebra arising from the dual of a compact
Lie group. We finally show, by example, that our
conjecture is false as stated, and raise the
question of whether there is a `good' class of
fusion algebras, which contains (a) the two 'good
classes' discussed above, namely, discrete groups
and compact group duals, and (b) only contains
fusion algebras for which the conjecture is
valid.
