  A gauge invariant mathematical formalism based on deformation quantization is
outlined to model an $\mathcal{N}=2$ supersymmetric system of a spin $1/2$
charged particle placed in a nocommutative plane under the influence of a
vertical uniform magnetic field. The noncommutative involutive algebra
$(C^{\infty}(\mathbb{R}^{2})[[\vartheta]],*^r)$ of formal power series in
$\vartheta$ with coefficients in the commutative ring
$C^{\infty}(\mathbb{R}^{2})$ was employed to construct the relevant
observables, viz., SUSY Hamiltonian $H$, supercharge operator $Q$ and its
adjoint $Q^{\dag}$ all belonging to the $2\times 2$ matrix algebra
$\mathcal{M}_{2}(C^{\infty}(\mathbb{R}^{2})[[\vartheta]],*^r)$ with the help of
a family of gauge-equivalent star products $*^{r}$. The energy eigenvalues of
the SUSY Hamiltonian all turned out to be independent of not only the gauge
parameter $r$ but also the noncommutativity parameter $\vartheta$. The
nontrivial Fermionic ground state was subsequently computed associated with the
zero energy which indicates that supersymmetry remains unbroken in all orders
of $\vartheta$. The Witten index for the noncommutative SUSY Landau problem
turns out to be $-1$ corroborating the fact that there is no broken
supersymmetry for the model we are considering.
