Considering that carbon nanotubes as a "rope" fiber would have extremely strong tensile strength but no compressive strength, the question then becomes: How do I use tensile strength to help avoid compression?
There are at least three ways to do this, two of which are already common in modern engineering.
1) Air supported structures - Imagine a giant, vertical dirigible. Now imagine many of these, long and narrow, used as a support structure. A carbon nanotube woven fabric could be made to withstand very high pressures within, creating a very rigid "beam" that is, strictly speaking, a balloon. (Side note, balloon animals for giants!)
2) Pretension - Similar to reinforced concrete, tension added to compression members before loading is already used to minimize flexing, allowing taller construction.
3) Active support structures - This one is my favorite, and incidentally, while requiring the most advanced technology, is capable of the tallest structures. Similar to the air-supported concept, active structures use a non-solid medium to keep a structure's form. Structural support provides thrust. That thrust is usually static, and it is exactly opposite the downward force of gravity. 333
Now, structural supports are not the only way to provide thrust - otherwise it would be very hard to get rockets off the ground. It's possible to use such an active system to support a building as well, but a rocket would not do, as the reaction mass is continuously lost. Instead, the idea for such a structure (commonly called a space fountain) is to force magnetically charged particulate matter in a stream through an evacuated tube, where they will be deflected back downward at the top through a return tube. That deflection provides the thrust to support the building. So far, this is irrelevant to the use of high tensile strength cables such as the carbon nanotube ropes. However, we would want our fountain tower to be rigid, to avoid our penthouse guests losing their lunch. Hmmmm, rigidity? This calls to mind the second method I described. Tension would allow for excess thrust to hold the tower at precisely the wanted height, and by using sufficiently angled stay lines to compensate for sideways thrust (from wind, earthquakes, etc.), just as on the mast of a ship, the tower would be held vertical. 513