Will there be friction? Is there friction between rough surfaces only (figure below)?

If we have two perfectly (mathematically) smooth surfaces, will there be friction between them (figure below)?

 A: If the two surfaces are real and perfectly smooth at an atomic level, and made of the same material, there will be chemical bonding between them.  They will fuse together as though they had always been the same object [1].  Even different metals can fuse together if left in contact for sufficient time, or if forced together, due to mutual diffusion of atoms across the interface [2].  It is only the presence of impurities, contaminants and irregularities in the surface structure which keeps them from fusing together.
[1]  https://en.wikipedia.org/wiki/Cold_welding - courtesy of CuriousOne
[2]  https://en.wikipedia.org/wiki/Diffusion_bonding 
A: Friction is a result of asperity-tops adhering together (as if they melted together) through chemical bonding. The more asperities that stick to the other surface, the larger the friction. To make them slide across each other, friction must be overcome, meaning that these "glued-together" tops must be broken.
I gave this explanation to another question not long ago.
If the surface is perfectly smooth, it is "one big asperity" that 100 % adheres to the other surface. Friction is here enourmous, and we no longer call it friction but instead cold-welding.
About perfect surfaces of extremely low friction, it is though possible to "trap" a perfectly uniform and extremely thin layer of air between the surfaces. They will Thus "float" over one another with practically no friction on this aerodynamic bearing mechanism. The air can in this way be trapped, because the surfaces being perfectly flat and parallel will cause vacuum if the air molecules try to escape, which keeps them in place.
If pushes hard enough together to squeeze out the air, the surfaces will be cold-welded together and theoretically impossible to separate again.
