On the Isomorphism of Semicircular Algebras

Building on OpenAI’s proof of isomorphism of free group factors, we give a self-contained proof of a corresponding result for $C^{\ast }$-algebras generated by free semicircular systems: the isomorphism class of a $C^{\ast }$-algebra generated by a semicircular $n$-tuple does not depend on $n\in \{2,3,\dots ,+\mathrm{\infty }\}$. Similarly, reduced free products of $n$ copies of the Jiang–Su algebra are isomorphic for different $n$ in this range. While reduced free group $C^{\ast }$-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 $C^{\ast }$- and $W^{\ast }$-algebras; a relative version of that property was previously used to derive free complementation results.