Positive Traces on Certain $\mathrm{S}\mathrm{L}(2)$ Coulomb Branches

For a noncommutative algebra $\mathcal{A}$ and an antilinear automorphism $\rho$ of $\mathcal{A}$, there is a notion of a positive trace. When we have a three-dimensional $\mathcal{N}=4$ gauge theory or four-dimensional $\mathcal{N}=2$ gauge theory compactified on a circle, classification of positive traces on its Coulomb branch $\mathcal{A}$ can give a better understanding of this theory. We classify positive traces on $\mathcal{A}$ in two cases. The first case is when $\mathcal{A}$ is a quantization of a Kleinian singularity of type $D$, with certain restriction on the quantization parameter. The second case is when $\mathcal{A}=K^{\mathrm{S}\mathrm{L}(2,\mathbb{C}[[t]])⋊{\mathbb{C}}_q^{\times }}({\mathrm{G}\mathrm{r}}_{{\mathrm{P}\mathrm{G}\mathrm{L}}_2})$ is an algebra containing $K$-theoretic Coulomb branches of pure $\mathrm{S}\mathrm{L}(2)$ and $\mathrm{P}\mathrm{G}\mathrm{L}(2)$ gauge theories.