On the Isomorphism of Semicircular Algebras

Abstract

Building on OpenAI's proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for $C^{*}$-algebras generated by free semicircular systems: the isomorphism class of a $C^{*}$-algebra generated by a semicircular $n$-tuple does not depend on $n\in\{2,3,\dots,+\infty\}$. Similarly, reduced free products of $n$ copies of the Jiang--Su algebra are isomorphic for different $n$ in this range. While reduced free group $C^{*}$-algebras of different ranks are not isomorphic (by a result of Pimsner--Voiculescu), we show that they have strong free absorption properties. At the heart of the proof is an automorphism approximation property for inclusions of $C^{*}$- and $W^{*}$-algebras; a relative version of that property was previously used to derive free complementation results.