Splitting carbon dioxide (CO2) into carbon and oxygen can in fact be accomplished, but there is a catch: doing so requires energy. If hydrocarbon fuels, which produce the greenhouse gas in the first place, supply that energy, thermodynamics tells us that the net result will be more CO2 than you started with.
Consider the proposal as a chemical reaction: CO2 plus energy yields carbon and oxygen. This formula essentially reverses coal combustion (carbon plus oxygen yields CO2 and energy). If energy from coal were applied to drive the decomposition reaction, more CO2 would be released than consumed, because no process is perfectly efficient.
Another option would be to harness a carbon-free energy source to drive a reaction that does not merely undo the combustion process but instead uses carbon dioxide as an input to generate useful, energy-rich products. At Sandia National Laboratories, we are working to apply concentrated sunlight to drive high-temperature thermal reactions that yield carbon monoxide, hydrogen and oxygen from CO2 and water. Carbon monoxide and hydrogen are basic chemical building blocks that find use in producing synthetic fuels, so we call this process “sunshine to petrol.”