A noncommutative 2-sphere generated by the quantum complex plane

TL;DR

This paper constructs a noncommutative 2-sphere by applying Woronowicz's C*-algebra theory to q-normal operators, linking the quantum complex plane with a quantum 2-sphere through crossed product algebras.

Contribution

It provides a novel construction of a quantum 2-sphere using the theory of C*-algebras generated by unbounded operators, specifically q-normal operators.

Findings

Computed the unique C*-algebra generated by a q-normal operator.

Described the algebra as a crossed product algebra.

Identified the quantum 2-sphere as a unitization of the algebra of functions vanishing at infinity.

Abstract

S. L. Woronowicz's theory of introducing C*-algebras generated by unbounded elements is applied to q-normal operators satisfying the defining relation of the quantum complex plane. The unique non-degenerate C*-algebra of bounded operators generated by a q-normal operator is computed and an abstract description is given by using crossed product algebras. If the spectrum of the modulus of the q-normal operator is the positive half line, this C*-algebra will be considered as the algebra of continuous functions on the quantum complex plane vanishing at infinity, and its unitization will be viewed as the algebra of continuous functions on a quantum 2-sphere.