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I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (1-q^2)$, where q $ \in (0,1)$. I need to find (the most simple) faithful representation of it but I couldn’t do it myself and the only reference I found is Pusz, Woronovicz “Twisted second quantization”. In this article I couldn’t understand a thing because it is written kind of with a physics notation.

Does someone by any chance know where else it is written? Or may be give me a hint how to construct it?

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Your algebra is a typical example of Wick *-algebra. Positive representations of general commutation relations allowing Wick ordering (Schmitt, Werner, Jorgensen) is a good reference for general theory of Wick *-algebras and their (bounded) representations. Example 1.2.3 is exactly what you need.

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  • $\begingroup$ Thank you, I figured out how does Fock representation of that work. But haven't you encountered to a faithful representations of TCCR such that generators don't tend to zero as $q$ tends to 1? I am not sure they are even exist... $\endgroup$
    – Invincible
    Commented Aug 17, 2019 at 12:57

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