Given a (finite) group $G$ and a subgroup $K$ we say that (G,K) is a Gelfand pair if the space of functions $f:G\to \mathbb{K}$ that are $K$-biinvariant (that means that $f(g)=f(k_1gk_2)$, for all $k_1,k_2 \in K$ is commutative. Where $\mathbb{K}$ is a agebraically closed field.
I would like to know if it make sense to speak in trivial Gelfand pairs, that is, groups where the space of all functions are already commutative. It seems that for instante $G=(\mathbb{Z}^2)^N$, for positive $N$, is an example of such group.