OAI.NuclearUltrapower.main_no_embedding

The theorem states that, working in universe level 0, there exist a nonzero unital complex C*-algebra A and a group homomorphism ι from G into the unitary group of A such that the following hold. Here G is the semidirect product of the additive group of 3-vectors over the dyadic rationals Z[1/2] by SL₃(ℤ) × ℤ, where a matrix acts linearly on the vectors and the integer n acts by multiplication by 2ⁿ. First, (A, ι) is the full group C*-algebra of G: the span of ι(G) is dense in A, and every homomorphism of G into the unitaries of a nonzero unital C*-algebra D extends uniquely to a unital -homomorphism A → D. Second, A is separable as a topological space. Third, for every nonzero unital C-algebra B that is nuclear in the tensor-norm sense (minimal and maximal C*-tensor norms agree on B ⊗ C for every C*-algebra C) and every free ultrafilter ω on ℕ (one containing no finite set), there is no injective unital -homomorphism from A into the norm ultrapower of B, namely bounded B-valued sequences modulo those tending to 0 along ω. Fourth, there exist a nonzero unital C-algebra O and two elements s₀, s₁ of O satisfying the Cuntz relations (each sᵢsᵢ = 1 and s₀s₀ + s₁s₁* = 1), with O universal for these relations, such that for every free ultrafilter ω there is likewise no injective unital *-homomorphism from A into the norm ultrapower of O. The theorem is admitted in the source, not proved.