Function algebras on the n-dimensional quantum complex space

TL;DR

The paper addresses constructing a topological C*-algebra for the noncommutative, noncompact, n-dimensional quantum complex space by classifying well-behaved representations of the coordinate algebra $O(C_q^n)$, realizing them as multiplication on $L^2$-spaces, and taking the norm-closure of a universal representation. The approach generalizes prior results for 1D and 2D quantum planes, using Woronowicz's framework of unbounded generators and spectral decomposition to obtain a complete representation theory that guides the C*-algebra construction. The main contributions are (i) a full classification of well-behaved *-representations (via a structured decomposition with $Q_k$ and $A_k$), (ii) an explicit realization on direct sums of $L^2$-spaces, and (iii) a concrete universal C*-algebra $C_0(C_q^n)$ generated by representations of a *-algebra whose classical limit separates points, delivering a noncommutative analogue of $C_0( olinebreak C^n)$. This framework provides a robust model for noncommutative function theory on quantum complex spaces and paves the way for further analysis of topological and geometric properties of $C_q^n$ via unbounded elements and spectral theory.

Abstract

The paper introduces a (universal) C*-algebra of continuous functions vanishing at infinity on the n-dimensional quantum complex space. To this end, the well-behaved Hilbert space representations of the defining relations are classified. Then these representations are realized by multiplication operators on an L2-space. The C*-algebra of continuous functions vanishing at infinity is defined by considering a *-algebra such that its classical counterpart separates the points of the n-dimensional complex space and by taking the operator norm closure of a universal representation of this algebra.

Theorems & Definitions (11)

• Proposition 1
• Lemma 2
• Lemma 3
• Definition 4
• Lemma 5
• proof
• Theorem 6
• proof
• Theorem 7
• proof