I would like to find a reference for the proof that functions of bounded variation make a Banach algebra. Same question for $BV\cap L^\infty$.

For the first part: wikipedia has a proof. So, I am just posting the link here. http://en.wikipedia.org/wiki/Bounded_variation#BV.28.CE.A9.29_is_a_Banach_algebra For the second part: are you considering functions of bounded variation on some interval? If so, then such a function can be written as the difference of two non decreasing functions, and hence is in $L^\infty$. So, this is answered by the first case. 


Since you ask for references, there are two readily traceable articles on algebras of functions of bounded variation in Manuacripta Mathematica by Blümlinger and Tichy. 

