Schubert Calculus and uniform property $\Gamma$

Abstract

We construct a simple, separable, unital, nuclear C$^*$-algebra without uniform property $\Gamma$. The construction is based on a new topological obstruction arising from the Thom-Porteous theory of degeneracy loci. Constructions of pathological nuclear C$^*$-algebras over the past 30 years have used Chern class calculations introduced by Villadsen to obstruct the existence of large trivial subbundles. Here, by contrast, we use determinantal Schur classes to force every bundle map between certain equal-rank vector bundles to vanish somewhere on the base space. A quadratic Schubert calculus computation shows that this obstruction can persist across an inductive system and ultimately obstructs the comparison of projections by traces in the uniform tracial completion. The relevant Thom-Porteous classes live in degree proportional to the square of the forced rank loss, which in turn forces dimension growth of the same order in the constituent homogeneous C$^*$-algebras of our example. This identifies a new geometric threshold in the structure theory of nuclear C$^*$-algebras, linking the presence or absence of uniform property $\Gamma$ to quadratic dimension growth.