Operator algebras with hyperarithmetic theory

摘要

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II1 factor R, L(Gamma) for Gamma a finitely generated group with solvable word problem, C* (Gamma) for Gamma a finitely presented group, C-lambda* (Gamma) for Gamma a finitely generated group with solvable word problem, C(2(omega)) and C(P) (where P is the pseudoarc). We also show that the Cuntz algebra O-2 has a hyperarithmetic theory provided that the Kirchberg embedding problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II1 factor (resp. C*-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II1 factors (resp. e.c. C*-algebras).