Publication Date:
2020-09-23
Description:
We show that the following operator algebras have hyperarithmetic theory: the hyperfinite II$_1$ factor $mathcal R$, $L(varGamma )$ for $varGamma $ a finitely generated group with solvable word problem, $C^*(varGamma )$ for $varGamma $ a finitely presented group, $C^*_lambda (varGamma )$ for $varGamma $ 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 problems have affirmative answers. Finally, we prove that if there is an existentially closed (e.c.) II$_1$ factor (resp. $extrm{C}^*$-algebra) that does not have hyperarithmetic theory, then there are continuum many theories of e.c. II$_1$ factors (resp. e.c. $extrm{C}^*$-algebras).
Print ISSN:
0955-792X
Electronic ISSN:
1465-363X
Topics:
Computer Science
,
Mathematics
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