OAI.KaplanskyConsequences.stable_finiteness_bundle

The theorem states that the proposition StableFinitenessBundle holds, namely the conjunction of three existence claims about C*-algebras. Throughout, a C*-algebra A is stably finite if for every n>0 and every n×n matrix v over A with vv=1 one has vv=1; it is properly infinite if there are s,t in A with ss=1, tt=1 and st=0; and a tracial state is a positive linear functional φ: A→ℂ with φ(1)=1 and φ(ab)=φ(ba). For a group G, a Hilbert space H and a star-homomorphism ρ from A into the bounded operators on H, the tensor algebra is the norm closure of the star algebra generated by the diagonal operators ρ(a) and the left-translation operators on ℓ²(G;H). (1) With F2 the free group on two generators and the reduced algebra the norm closure of the star algebra generated by left translations on ℓ²(F2;ℂ), this algebra is C-simple (every closed two-sided star ideal is zero or everything) and stably finite, and there is a nontrivial C*-algebra M that is C*-simple and stably finite, with an injective star-homomorphism ρ into the bounded operators on some complex Hilbert space H, such that the F2 tensor algebra of ρ is properly infinite. (2) There is a nontrivial separable C*-algebra B with a normalized 1-quasitrace τ (τ(1)=1) that extends to a 1-quasitrace on 2×2 matrices over B through the upper-left corner, and an injective star-homomorphism ρ of B into the bounded operators on some Hilbert space H, such that the F2 tensor algebra of ρ is properly infinite and has no normalized 1-quasitrace with that 2×2 extension property. (3) There is a nontrivial separable stably finite C*-algebra with no tracial state. Here a 1-quasitrace is a map τ: A→ℂ with τ(xx) a nonnegative number, τ(xx)=τ(xx*), additivity of the form τ(h+ik)=τ(h)+iτ(k) for self-adjoint h and k, and linearity on every closed commutative star-subalgebra.