  Following [Beem C., Peelaers W., Rastelli L., Comm. Math. Phys. 354 (2017),
345-392, arXiv:1601.05378] and [Etingof P., Stryker D., SIGMA 16 (2020), 014,
28 pages, arXiv:1909.13588], we undertake a detailed study of twisted traces on
quantizations of Kleinian singularities of type $A_{n-1}$. In particular, we
give explicit integral formulas for these traces and use them to determine when
a trace defines a positive Hermitian form on the corresponding algebra. This
leads to a classification of unitary short star-products for such
quantizations, a problem posed by Beem, Peelaers and Rastelli in connection
with 3-dimensional superconformal field theory. In particular, we confirm their
conjecture that for $n\le 4$ a unitary short star-product is unique and compute
its parameter as a function of the quantization parameters, giving exact
formulas for the numerical functions by Beem, Peelaers and Rastelli. If $n=2$,
this, in particular, recovers the theory of unitary spherical Harish-Chandra
bimodules for ${\mathfrak{sl}}_2$. Thus the results of this paper may be viewed
as a starting point for a generalization of the theory of unitary
Harish-Chandra bimodules over enveloping algebras of reductive Lie algebras
[Vogan Jr. D.A., Annals of Mathematics Studies, Vol. 118, Princeton University
Press, Princeton, NJ, 1987] to more general quantum algebras. Finally, we
derive recurrences to compute the coefficients of short star-products
corresponding to twisted traces, which are generalizations of discrete
Painlev\'e systems.
