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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. AHermitian 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 thispaper we classify positive definite invariant Hermitian forms on generalized$q$-Weyl algebras.

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