Central extensions and almost representations
Marius Dadarlat, Forrest Glebe

TL;DR
This paper constructs a central extension of unitary groups associated with sequences of tracial C*-algebras, using it to identify obstructions to approximating asymptotic group homomorphisms by true homomorphisms, with applications to stability properties.
Contribution
It introduces a canonical central extension framework for asymptotic homomorphisms, linking cohomology obstructions to stability and approximation problems in operator algebras.
Findings
Obstruction classes in cohomology prevent perturbation of asymptotic homomorphisms to genuine homomorphisms.
The pairing of cohomology classes yields numerical invariants related to winding numbers.
The full group C*-algebra is not C*-stable if its second cohomology is non-zero.
Abstract
For a sequence of unital tracial -algebras we construct a canonical central extension of the unitary group by using de la Harpe-Skandalis pre-determinant. For an asymptotic group homomorphism the corresponding pullback of the canonical central extension gives a 2-cohomology class in which obstructs the perturbation of to a sequence of true homomorphisms of groups . The pairing of the obstruction class with elements of yields numerical invariants in that subsume the winding number invariants of Kazhdan, Exel and Loring. For generality, we allow bounded asymptotic homomorphisms to map the group into the…
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Taxonomy
TopicsRings, Modules, and Algebras
