$LiMn_2O_4$ catalyst for $CO_2$ decomposition was synthesized by oxidation method for 30 min at 600$^{\circ}C$ in an electric furnace under air condition using manganese(II) nitrate $(Mn(NO_3)_2{\cdot}6H_2O)$, Lithium nitrate ($LiNO_3$) and Urea $(CO(NH_2)_2)$. The synthesized catalyst was reduced by $H_2$ at various temperatures for 3 hr. The reduction degree of the reduced catalysts were measured using the TGA. And then $CO_2$ decomposition rate was measured using the reduced catalysts. Phase-transitions of the catalysts were observed after $CO_2$ decomposition reaction at an optimal decomposition temperature. As the result of X-ray powder diffraction analysis, the synthesized catalyst was confirmed that the catalyst has the spinel structure, and also confirmed that when it was reduced by $H_2$, the phase of $LiMn_2O_4$ catalyst was transformed into $Li_2MnO_3$ and $Li_{1-2{\delta}}Mn_{2-{\delta}}O_{4-3{\delta}-{\delta}'}$ of tetragonal spinel phase. After $CO_2$ decomposition reaction, it was confirmed that the peak of $LiMn_2O_4$ of spinel phase. The optimal reduction temperature of the catalyst with $H_2$ was confirmed to be 450$^{\circ}C$(maximum weight-increasing ratio 9.47%) in the case of $LiMn_2O_4$ through the TGA analysis. Decomposition rate(%) using the $LiMn_2O_4$ catalyst showed the 67%. The crystal structure of the synthesized $LiMn_2O_4$ observed with a scanning electron microscope(SEM) shows cubic form. After reduction, $LiMn_2O_4$ catalyst became condensed each other to form interface. It was confirmed that after $CO_2$ decomposition, crystal structure of $LiMn_2O_4$ catalyst showed that its particle grew up more than that of reduction. Phase-transition by reduction and $CO_2$ decomposition ; $Li_2MnO_3$ and $Li_{1-2{\delta}}Mn_{2-{\delta}}O_{4-3{\delta}-{\delta}'}$ of tetragonal spinel phase at the first time of $CO_2$ decomposition appear like the same as the above contents. Phase-transition at $2{\sim}5$ time ; $Li_2MnO_3$ and $Li_{1-2{\delta}}Mn_{2-{\delta}}O_{4-3{\delta}-{\delta}'}$ of tetragonal spinel phase by reduction and $LiMn_2O_4$ of spinel phase after $CO_2$ decomposition appear like the same as the first time case. The result of the TGA analysis by catalyst reduction ; The first time, weight of reduced catalyst increased by 9.47%, for 2${\sim}$5 times, weight of reduced catalyst increased by average 2.3% But, in any time, there is little difference in the decomposition ratio of $CO_2$. That is to say, at the first time, it showed 67% in $CO_2$ decomposition rate and after 5 times reaction of $CO_2$ decomposition, it showed 67% nearly the same as the first time.