The field of quantum $GL(N,\mathbb{C})$ in the C$^*$-algebraic setting

Given a unital $*$-algebra $\mathscr{A}$ together with a suitable positive filtration of its set of irreducible bounded representations, one can construct a C$^*$-algebra $A_0$ with a dense two-sided ideal $A_c$ such that $\mathscr{A}$ maps into the multiplier algebra of $A_c$. When the filtration is induced from a central element in $\mathscr{A}$, we say that $\mathscr{A}$ is an s$^*$-algebra. We also introduce the notion of $\mathscr{R}$-algebra relative to a commutative s$^*$-algebra $\mathscr{R}$, and of Hopf $\mathscr{R}$-algebra. We formulate conditions such that the completion of a Hopf $\mathscr{R}$-algebra gives rise to a continuous field of Hopf C$^*$-algebras over the spectrum of $R_0$. We apply the general theory to the case of quantum $GL(N,\mathbb{C})$ as constructed from the FRT-formalism.