A short note on strong convergence of $q$-Gaussians

In this note, we prove strong convergence of $q$-Gaussians with respect to a parameter $q$, which implies the spectrum of any self-adjoint non-commutative polynomial in $q$-Gaussians is continuously deformed with respect to $q$. With Bożejko’s Haagerup-type inequality, we follow Brannan’s approach to proving the asymptotic strong convergence of the free orthogonal quantum group.