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

TL;DR

This paper develops a C*-algebraic framework for quantum groups, specifically quantum $GL(N,\, ext{C})$, by introducing new algebraic structures and conditions for continuous fields of Hopf C*-algebras.

Contribution

It introduces the notions of s*-algebras, R-algebras, and Hopf R-algebras, and applies these to construct a C*-algebraic model of quantum $GL(N,\, ext{C})$.

Findings

Constructed a C*-algebraic model for quantum $GL(N,\, ext{C})$.

Established conditions for continuous fields of Hopf C*-algebras.

Developed a new algebraic framework for quantum groups in the C*-algebra setting.

Abstract

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.