Starting from the other end of things, it clearly is possible to synthesise organic compounds from atmospheric concentrations of CO2 as nature does it already, the question is what sort of yield you can get.
My understanding is that using algae to do this via photosynthesis is limited by the fact that with wild strains you need to harvest and extract the biomass before you can get at the fuel, and you need to spread things out in a very thin film to get enough sunlight. Using modified strains helps solve the first problem, using (more efficient) synthetic photovoltaics, or another power source, with a process like this potentially helps solve the second.
To respond to another reply to this post, I don't think complex structures that maximise surface area would be necessary - you could just bubble air through the tank. As mentioned above, bubbling CO2 rich waste gasses from a power plant would likely be even more efficient.
My understanding is that using algae to do this via photosynthesis is limited by the fact that with wild strains you need to harvest and extract the biomass before you can get at the fuel, and you need to spread things out in a very thin film to get enough sunlight. Using modified strains helps solve the first problem, using (more efficient) synthetic photovoltaics, or another power source, with a process like this potentially helps solve the second.
To respond to another reply to this post, I don't think complex structures that maximise surface area would be necessary - you could just bubble air through the tank. As mentioned above, bubbling CO2 rich waste gasses from a power plant would likely be even more efficient.