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For a compact Lie group $G$ we consider a lattice gauge model given by the$G$-Hamiltonian system which consists of the cotangent bundle of a power of $G$with its canonical symplectic structure and standard moment map. We explicitlyconstruct a Fedosov quantization of the underlying symplectic manifold usingthe Levi-Civita connection of the Killing metric on $G$. We then explain andrefine quantized homological reduction for the construction of a star producton the symplectically reduced space in the singular case. Afterwards we showthat for $G = \operatorname{SU} (2)$ the main hypotheses ensuring the method ofquantized homological reduction to be applicable hold in the case of ourlattice gauge model. For that case, this implies that the - in general singular- symplectically reduced phase space of the corresponding lattice gauge modelcarries a star product.

Deformation quantization and homological reduction of a lattice gauge model