It would have to be pressure.
Solid C02 is called "dry ice" because it goes straight to gas when it melts at atmospheric pressure.

I think more work should be done with algae.

Totally agree about algae.

An experiment is under way at the Danish continental shelf, at a non-negligible scale.
Smaller trials were made in deserts, using man-made containers.
As a variants, they can be fed directly with the CO2 produced by a lime plant or a power plant.

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Liquid: agree again. 1 bar is too little for liquid CO2, exists only as a solid or a gas at that pressure.

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Trees turning CO2 into O2: unfortunately, a tree doesn't do it forever. To understand the CO2 question, the trick is to consider what happens to carbon, as CO2->O2 means C must land somewhere.

Plants and trees capture CO2 as long as they use C to grow new vegetable material. When the tree (or its leafs, hence the seasonal cycles in atmospheric CO2) dies, its stuff rots and the contained C is largely released in the atmosphere, as CH4 (powerful greenhouse gas but oxidized in CO2 after some time) or CO2.

In other words, to absorb atmospheric CO2, a forest isn't enough. You need a growing forest - which isn't really the case right now. Or maybe bury all dead trees, but the scale is impractical.

This is why people speak of "carbon wells" and other idiomatics. The very point is to sequester carbon, be it CO2 or C or CaCO3 or algae.
Now, consider that all coal, oil, gas we're burning in 100-200 years is the result of CO2 absorption over 100 million years from a time where Earth had more CO2 than O2. Extract and burn all these fossil energies, you (we) go back to the atmosphere's composition then. Drawback: humans can't breathe it.

This puts an absolute upper limit to the amount of fossil energies mankind may burn. Scientific consensus tells much lower limits exist because of climate.

As you see, reducing the amount of fossil energy we use is not a solution. At some point (maybe already behind us...) we'll have to completely give up fossil energies and even capture back the CO2 already added to the atmosphere. My personal opinion is that any effort to reduce the consumption is an effort diverted from the only useful goal, which is stop using fossil energies. Solutions are completely different in both cases.

The main obstacle to fighting against greenhouse effect is that [B]its timescale isn't understandable for politicians[B]. Anything over 6 months is abstract for them. Tell them "20 years", they understand "not me".

I don't suppose it is possible to do what a growing plant does by way of some machine?

I agree. Yes, we need to get energy from something other than fossil fuels. We need more trees to grow.

You mean at a purely chemical reaction level?
I think algae could be genetically modified to be as simple as possible.

Well, yes, there have been and are attempts to convert sunlight in chemical energy without vegetables.

Some use heat to crack CO2 in CO and O2, then CO reacts with other molecules to make fuels, chemicals... like methanol and bigger ones. Cracking H2O is also possible but more difficult. This is a way with reasonable costs and conversion efficiency.

Others use photochemical methods. The worst ones have photocells tightly coupled with electrochemical cells - not significantly better than a loose coupling for the bad efficiency. The best ones try to induce directly photochemical reactions, but as far as I know, Nature leads technology for this.

The basic issue is, as usual for renewables, cost. Though machines would need less surface than vegetables for the same power, vegetables are way cheaper. Vegetables oil or corn ethanol without fuel taxes are cheaper than gasoline with fuel taxes - nothing similar with today's machines.

As for the surface needed, today's biofuels consume good arable land, but current attempts (in a good way to succeed) use areas of less value to Mankind, hence the experiments with algae in the ocean or in containers in a desert.

Algae have advantages over terrestrial vegetables. They accept seawater, and they grow faster with the same amount of light.

Monocellular photosynthesizers are also investigated. Some can be modified to produce hydrogen directly - but I wouldn't have anything against methane or ethanol or tar.

CO2 has uses, unfortunately, we make more of it than we need.

The upside is that it's fairly cooperative in terms of means of disposal or sequestration. It liquifies at fairly reasonable pressures at room temperature, making it a good fire extinguisher. CO2 fire extinguishers are filled with liquid CO2 under pressure. It also dissolves in water, making soda water (carbonic acid) which is the carbonation in soft drinks. As it's already been pointed out, we exhale it, so it isn't a dangerous toxin, just something we need to exhale. Growing plants need it to live, so the right balance of growing plants and fossil fuels should be achievable, but for the fact that the majority of agriculture in third world countries is based on the "slash and burn" method of creating fertile land for planting food crops. A lot of the atmosphere's CO2 comes from "slash and burn" farming, and so long as the population continues to spiral out-of-control in third-world countries, we'll have that problem.

What to do? Power plants that burn coal can sequester the CO2 fairly easily in underground "sinks". Take an oil well that isn't producing much oil anymore and pump CO2 down it, and the oil well starts producing again. If the well passes through calcium-bearing minerals, they can react with the CO2, stabilizing it for centuries (or longer).

But there's a simpler approach that avoids CO2 completely. Methane is 23x as powerful a greenhouse gas as CO2 is, and no one (that I know of) is trying to do anything about sequestering methane. Recognize that all you need to do to halt methane's greenhouse effects is burn it, making CO2 (which reduces its greenhouse effect by a factor of 22). I've read that the current greenhouse crisis is due to there being 3% more CO2 than there should be to keep the plant in thermal equilibrium. It's a whole lot easier to reduce the methane released into the atmosphere by 3%/23 (= 0.13%) than it is to reduce the CO2 by 3%. Burning it is the most-obvious solution.

Sequestration of CO2 down a well should also be fairly simple, if there's water in the well, at some point the pressure at some depth should be enough to turn the CO2 liquid, like it is in those fire extinguishers, but as with anything, it's a matter of cost and political commitment more than technological ignorance holding us back.

The project I'm currently working on is a solar thermal power plant, which is one of those big "mirror farms" that focus all the sunlight at a central receiver, and uses the concentrated solar power to generate electricity via conventional steam-turbine/generator(s). I have a vested interest in promoting that approach, but it should also be noted that there is enough geothermal energy available to generate hundreds of times as much electricity as we need, and they're fairly simple to build, too.

Electric cars can be built that would be practical, and I've just (earlier this week) submitted an idea for making cars with 30HP motors perform like they have V-8s in them to my company, so I can't go further into it without p*ssing-off my company's management (if they like the idea). No "rocket science" necessary to make it work, just innovative thinking.

One of the simplest ideas I've run across for fixing global warming is for commercial buildings to paint their roofs white. A white roof will tend to reflect the incident solar energy back into space, instead of making the building hot, requiring more A/C to keep it cool in the summertime, increasing the demand for electricity and the demand for fossil fuels at electric plants.

In some parts of the country, combining a heat pump with an underground water tank (heat sink/source) is a cost-effective means of providing cool air during hot weather and warm air during cold weather.

There are lots of good, practical ideas out there, the real problem (as it's been pointed out) is that they require commitment from politicians, and politicians seldom think about long-term problems.

Let it boil. Extra atmospheric Co2 will ensure a warmer planet which will support a greater human population.