On the Fock representation of the q-commutation relations

TL;DR

This paper investigates the structure of the C*-algebra generated by q-commutation relations on twisted Fock space, revealing a canonical unitary transformation linking it to the full Fock space and identifying conditions for algebraic equality.

Contribution

It introduces a canonical unitary that maps the q-commutation algebra to the full Fock space and establishes when the algebra contains the Cuntz algebra R^0.

Findings

Existence of a canonical unitary U_q linking R^q to the full Fock space.

Containment of the Cuntz algebra R^0 within the transformed algebra.

Equality of the algebras for |q|<0.44.

Abstract

The q-commutation relations in the title are those that have recently received much attention, and that for -1<q<1 provide an interpolation between Bosonic and Fermionic statistics, passing through free statistics at q=0. We look at the C*-algebra R^q generated by the representation of these relations on the twisted Fock space of Bozejko-Speicher, and we find a canonical unitary U_q from the twisted Fock space to usual full Fock space, such that U_q R^q (U_q)^* contains the Cuntz algebra R^0, and such that we have equality for |q|<0.44.