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

Abstract

Given a unital $\ast$-algebra $\mathcal{A}$ together with a suitable positive filtration of its set of irreducible bounded representations, one can construct a C${}^{\ast }$-algebra $A_0$ with a dense two-sided ideal $A_c$ such that $\mathcal{A}$ maps into the multiplier algebra of $A_c$. When the filtration is induced from a central element in $\mathcal{A}$, we say that $\mathcal{A}$ is an s${}^{\ast }$-algebra. We also introduce the notion of $\mathcal{R}$-algebra relative to a commutative s${}^{\ast }$-algebra $\mathcal{R}$, and of Hopf $\mathcal{R}$-algebra. We formulate conditions such that the completion of a Hopf $\mathcal{R}$-algebra gives rise to a continuous field of Hopf C${}^{\ast }$-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.

41 pages

References in corpus (1)

• K-homology class of the Dirac operator on a compact quantum group

Cited by in corpus (1)

• Multiparameter quantum Cauchy-Binet formulas