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It was shown by Pimsner and Voiculescu in 1982 that the reduced group $C^{*}$-algebras $C^{*}_{r}(\mathbb{F}_{n})$ and $C^{*}_{r}(\mathbb{F}_{m})$ are isomorphic if and only if $n = m$ (here, $\mathbb{F}_{n}$ denotes the free group on $n$ generators). The proof of this fact relies on the computation of the $K_{1}$ groups of both algebras, which coincide if and only if $n = m$.

I suspect that if $(x_{1}, x_{2}, \dots)$ forms a free semicircular family with respect to a trace $\tau$, then $C^{*}(x_{1}, \dots, x_{n}, \tau) \cong C^{*}(x_{1}, \dots, x_{m}, \tau)$ if and only if $n = m$. Certainly any proof of this statement can not rely on the computation of $K$ groups as they are isomorphic. I am wondering if anyone knows a good reference for this (assuming it is true).

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    $\begingroup$ I just spoke with Ken Dykema, and he informed me that the (non)isomorphism of the $C^*$-algebras generated by different-sized semicircular systems is in fact an open problem! $\endgroup$ Feb 28, 2014 at 7:10

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