Hydrogen floats and escapes
Hydrogen cannot be contained in the atmosphere, because it will float to the top, and then escape the Earth.
Already today we can break up carbon dioxide. There are several ways of doing it. For instance we can pewpew it with a laser to mention one of the more exotic examples.
But the major problem with breaking up carbon dioxide is that all of these processes require energy. And the excess $CO_2$ that we have a problem with now was created when we were generating energy. So in order to split $CO_2$, we need to generate at least as much energy as was gained when the $CO_2$ was formed in the first place, and this time without having any $CO_2$ as a waste product.
This of course means that we should just skip the $CO_2$-generating step all-together and just use the carbon dioxide free energy directly instead.
So to answer your question: what (future) technology can be used to break up carbon dioxide? Well you have a buffet to choose from already.
The break-through you are looking for is a method of generating carbon dioxide free energy in such abundance that we can cover all our energy consumption needs, and have enough left over to start breaking up carbon dioxide too.
EDIT: The one technology that would be really cool and futuristic in this context — that would break up carbondioxide using sunlight as its source of energy — is one that emulates photosynthesis, in other words: artificial photosynthesis. If humanity can get artificial photosynthesis to work, then we will have a very cool technology indeed, a cool technology that will open the door to many solutions to woes that we face today.
EDIT by OP: This technology could be used as an decentralized energy source. For it to work it requires further advances in "light harvesting, charge separation, catalysis, semiconductors, nanotechnology, modelling from synthetic biology and genetic engineering, photochemistry and photophysics, photoelectrochemistry, catalysis, reaction mechanisms and device engineering."