Longitudinal Double-Spin Asymmetry A$_{\mathrm{LL}}$ for Inclusive Jet Production in Polarized Proton Collisions at $\surd $s $=$ 510 GeV

The STAR experiment at the Relativistic Heavy-Ion Collider (RHIC) at Brookhaven National Laboratory is carrying out a spin physics program in high-energy polarized proton collisions at center of mass energies up to 510 GeV, to gain a deeper insight into the spin structure and dynamics of the proton. The polarized gluon distribution function can be constrained in longitudinally polarized proton collisions through jet / di-jet production. Recent global analyses, which include results of the measurement of A$_{\mathrm{LL}}$ for inclusive jet production at 200 GeV at mid-rapidity at the STAR experiment, provide evidence of a non-zero gluon polarization in the measured range of partonic momentum fraction~of approximately 0.05\textless x\textless 0.3. We present preliminary results of the measurement of A$_{\mathrm{LL}}$ for inclusive jet production at 510 GeV. The higher center-of-mass energy allows to probe the polarized gluon distribution function at smaller x values. The data were recorded at the STAR experiment during the 2013 running period at mid-rapidity (\textbar $\eta $\textbar \textless 0.9). The integrated luminosity recorded at STAR used for this study is approximately 250 pb$^{\mathrm{-1}}$. We discuss and compare these results with previous STAR results and various next-to-leading order (NLO) calculations.