  The universal $R$ operator for the positive representations of split real
quantum groups is computed, generalizing the formula of compact quantum groups
$U_q(g)$ by Kirillov-Reshetikhin and Levendorski\u{\i}-Soibelman, and the
formula in the case of $U_{q\tilde{q}}(sl(2,R))$ by Faddeev, Kashaev and
Bytsko-Teschner. Several new functional relations of the quantum dilogarithm
are obtained, generalizing the quantum exponential relations and the pentagon
relations. The quantum Weyl element and Lusztig's isomorphism in the positive
setting are also studied in detail. Finally we introduce a $C^*$-algebraic
version of the split real quantum group in the language of multiplier Hopf
algebras, and consequently the definition of $R$ is made rigorous as the
canonical element of the Drinfeld's double $\textbf{U}$ of certain multiplier
Hopf algebra $\textbf{Ub}$. Moreover a ribbon structure is introduced for an
extension of $\textbf{U}$.
