A counterexample to the Global Glimm Problem

We construct a unital, separable, nuclear C*-algebra whose hereditary subalgebras have no nonzero finite-dimensional representations, but which has no full square-zero element. This gives negative answers to the Global Glimm Problem, originating in Kirchberg and Rørdam’s work, and to Elliott and Rørdam’s full matrix-cone question. The algebra is a continuous field of CAR algebras over a countable product of finite quaternionic projective spaces. Every selfadjoint element is scalar in some fiber; consequently, there are no two orthogonal full positive elements and no full soft positive element. The construction realizes Dranishnikov–Levin’s equivariant obstruction in quaternionic bundles within the CAR-field framework of Hirshberg, Rørdam, and Winter.

GitHub repository: eulogica/global-glimm-counterexample