For a noncommutative algebra $A$ and an antilinear automorphism $\rho$ of $A$ there is a notion of a positive trace. Positive traces appear in study of supersymmetric gauge theories. We classify positive traces on $A$ in two cases. The first case is when $A$ is a quantization of a Kleinian singularity of type $D$, with certain restriction on the quantization parameter. The second case is when $A=K^{ \operatorname{ SL}(2,\mathbb{ C}[[t]])\rtimes \mathbb{ C}_q^{ \times}}(\operatorname{ Gr}_{ \operatorname{ PGL}_2})$ is an algebra containing $K$-theoretic Coulomb branches of pure $\operatorname{ SL}(2)$ and $\operatorname{ PGL}(2)$ gauge theories.
Categories: hep-th math-ph math.MP math.RT