Operator algebras with hyperarithmetic theory

We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II${}_1$ factor $\mathcal{R}$, $L(\mathrm{\Gamma })$ for $\mathrm{\Gamma }$ a finitely generated group with solvable word problem, $C^{\ast }(\mathrm{\Gamma })$ for $\mathrm{\Gamma }$ a finitely presented group, $C_{\lambda }^{\ast }(\mathrm{\Gamma })$ for $\mathrm{\Gamma }$ 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${}^{\ast }$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II${}_1$ factors (resp. e.c. C${}^{\ast }$-algebras).