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Given a unital $*$-algebra $\mathscr{A}$ together with a suitable positivefiltration of its set of irreducible bounded representations, one can constructa 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 isinduced from a central element in $\mathscr{A}$, we say that $\mathscr{A}$ isan s$^*$-algebra. We also introduce the notion of $\mathscr{R}$-algebrarelative to a commutative s$^*$-algebra $\mathscr{R}$, and of Hopf$\mathscr{R}$-algebra. We formulate conditions such that the completion of aHopf $\mathscr{R}$-algebra gives rise to a continuous field of HopfC$^*$-algebras over the spectrum of $R_0$. We apply the general theory to thecase of quantum $GL(N,\mathbb{C})$ as constructed from the FRT-formalism.

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