Operator algebras with hyperarithmetic theory

Abstract: We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$1$ factor $\mathcal R$, $L(\Gamma)$ 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 $C(\mathbb P)$ (where $\mathbb P$ is the pseudoarc). We also show that the Cuntz algebra $\mathcal O_2$ has a hyperarithmetic theory provided that the Kirchberg embedding problem has an affirmative answer. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. C$^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. C$^*$-algebras).