Twisted Traces and Positive Forms on Generalized $q$-Weyl Algebras

Let $\mathcal{A}$ be a generalized $q$-Weyl algebra, it is generated by $u$, $v$, $Z$, $Z^{-1}$ with relations $ZuZ^{-1}=q^2u$, $ZvZ^{-1}=q^{-2}v$, $uv=P\big(q^{-1}Z\big)$, $vu=P(qZ)$, where $P$ is a Laurent polynomial. A Hermitian form $(\cdot,\cdot)$ on $\mathcal{A}$ is called invariant if $(Za,b)=\big(a,bZ^{-1}\big)$, $(ua,b)=(a,sbv)$, $(va,b)=\big(a,s^{-1}bu\big)$ for some $s\in \mathbb{C}$ with $|s|=1$ and all $a,b\in \mathcal{A}$. In this paper we classify positive definite invariant Hermitian forms on generalized $q$-Weyl algebras.