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=1/4μ1+3/4μ2, where μ1 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.
Related
- Null polynomials over a finite ring need not form a two-sided ideal
- Dual Area-Volume-of-Fluid (DAVOF): co-advecting volume and face-area fractions for a closed-form, cell-local interface normal and position on unstructured meshes
- The hyperfinite II_1 factor is not quasidiagonal
- A quasidiagonal C*-algebra with a nonquasidiagonal maximal tensor square
- Exotic knot traces and the epsilon-invariant
BibTeX
@misc{moradi2026quasidiagonal,
title = {{Quasidiagonal traces need not form a face}},
author = {Mehdi Moradi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18793},
eprint = {2609.18793},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18793},
}
RIS
TY - UNPB
TI - Quasidiagonal traces need not form a face
AU - Mehdi Moradi
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18793
UR - https://arxiv.org/abs/2609.18793
AB - 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=1/4μ_1+3/4μ_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.
ER -