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We found an explicit construction of a representation of the positive quantumgroup $GL_q^+(N,\R)$ and its modular double $GL_{q\til[q]}^+(N,\R)$ by positiveessentially self-adjoint operators. Generalizing Lusztig's parametrization, wefound a Gauss type decomposition for the totally positive quantum group$GL_q^+(N,\R)$ for $|q|=1$, parametrized by the standard decomposition of thelongest element $w_0\in W=S_{N-1}$. Under this parametrization, we foundexplicitly the relations between the standard quantum variables, the relationsbetween the quantum cluster variables, and realizing them using non-compactgenerators of the $q$-tori $uv=q^2 vu$ by positive essentially self-adjointoperators. The modular double arises naturally from the transcendentalrelations, and an $L^2(GL_{q\til[q]}^+(N,\R))$ space in the von Neumann settingcan also be defined.

Gauss-Lusztig Decomposition for $GL_q^+(N,R)$ and Representation by q-Tori