Trace-norm rigidity for reduced products of unitary groups and matrix algebras

Abstract

We study homomorphisms, with a focus on isomorphisms, between the tracial metric reduced products of finite dimensional unitary groups and of matrix algebras. A variant of Ulam stability for unitary groups and a classification of the almost surjective continuous homomorphisms between finite dimensional unitary groups are proved and then used to show that all isomorphisms of product form of these tracial reduced products are induced by almost permutations of the coordinates and coordinatewise application of automorphisms. We prove coordinate recognition for these reduced products and obtain under set theoretic assumptions rigidity and classification results for their full automorphism groups. For tracial reduced matrix algebras we obtain such rigidity result in the more general context of center-preserving $*$-homomorphisms.

Disclosure

“arch Foundation) under Germany’s Excellence Strategy EXC 2044/2 -390685587, Mathematics Münster: Dynamics–Geometry–Structure. Andreas Thom acknowledges funding by the Deutsche Forschungsgemeinschaft (SPP 2026 “Geometry at infinity”). ChatGPT 5.5 was used to assist in drafting parts of this manuscript. All content was reviewed and substantially revised by the authors, who are responsible for the final text. References [1] Vadim Aleks”

PDF page 26

Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-23T19:01:36.156702Z

Structural counts