Quasidiagonal traces need not form a face

Abstract

Let \(G\) be an infinite residually finite countable discrete group with Kazhdan's property~\((T)\). Assume that its finite-dimensional irreducible unitary representations admit an exhaustive ordering with nondecreasing dimensions tending to infinity and bounded consecutive dimension ratios. From this data we construct a separable unital residually finite-dimensional \(C^*\)-algebra with a faithful quasidiagonal tracial state \(τ=\frac14μ_1+\frac34μ_2\), where \(μ_1\) is not quasidiagonal. For the resulting algebra, quasidiagonal traces do not form a face of the tracial state space. The construction has two copies of a weighted representation ladder joined at their top levels. We prove that every quasidiagonal trace gives equal weight to the two copies. The proof combines uniform Kazhdan spectral projections, exact ranks of rounded finite-rank compressions, and a weighted Hilbert--Schmidt comparison that controls all representation blocks.