  We study the bounded endomorphisms of $\ell_{N}^2(G)=\ell^2(G)\times \dots
\times\ell^2(G)$ that commute with translations, where $G$ is a discrete
abelian group. It is shown that they form a C*-algebra isomorphic to the
C*-algebra of $N\times N$ matrices with entries in $L^\infty(\widehat{G})$,
where $\widehat{G}$ is the dual space of $G$. Characterizations of when these
endomorphisms are invertible, and expressions for their norms and for the norms
of their inverses, are given. These results allow us to study Riesz systems
that arise from the action of $ G $ on a finite set of elements of a Hilbert
space.
