White Walls and White Backgrounds are often a challenge. Here are some tips.
I imagine some walls in the real world aren't as "White" as we think they are. For instance, if we stood in front of the white wall with a very bright flashlight, would the beam of the flashlight be even "whiter". If so, then the wall wasn't really "white", but rather we perceived it as whiter than it was.
(Mathematically, we can't make a surface any whiter then RGB 255,255,255. - So if a wall s not 100% fully illuminated, then we have to make it a little less white (gray) in order to allow more whitefor other objects. This is what makes some scenes too grey.
Here is a simple room with glossy "white" walls.
The walls are pretty "gray". However, even if they were "white", we would expect the lamp shade to be a brighter white. We can make the whole scene brighter with the brightness slider at the top of the rendering.
Here I increased overall brightness by 10%.
That made the walls "whiter", but if I ran brightness all the way up, the rest of the scene would wash out.
Another way to make a single object brighter is to apply self glow to the material on the surface. Here I placed a face on the wall with the same white material, but gave it a 20% self glow. This make the rectangle more white, as if it had more light shining on it. (Self glow is avaiable on the glow tab of the material editor. Note: Self glow makes the surface brighter, but does not treat it a s light source)
Here I applied a 15% self glow to both walls. This made the walls lighter, but the automatic "photometric" lighting adjuster darkened the rest of the scene to compensate for the bright walls. (This can be turned off by locking the Automatic Luminance .)
So finally, I adjusted the brightness slider at the top of the screen to make the whole image brighter:
I think that with a combination of self glow and the brightness slider you may be able to get close to the effect you are looking for.
Note: Lighting channels will make this easier - their is one lighting channel which adjusts the self glow with a slider bar, so you can adjust self glow and overall brightness after the rendering is complete. See also: Lighting Channels