A groupoid approach to interacting fermions

We consider the algebra $dot{mathfrak H}(mathcal L)$ of inner-limit derivations over the ${rm GICAR}$ algebra of a fermion gas populating an aperiodic Delone set $mathcal L$. Under standard physical assumptions such as finite interaction range, Galilean invariance and continuity with respect to $mathcal L$, we demonstrate that $dot{mathfrak H}(mathcal L)$ can be completed to a groupoid-solvable pro-$C^ast$-algebra. Our result is the first step towards unlocking the $K$-theoretic tools available for separable $C^ast$-algebras for applications in the context of interacting fermions.