Suppose that \({\mathcal L}_n\) is a noncommutative analytic Toeplitz algebra defined by the left regular representation of the free semigroup \({\mathcal F}_n\) on \(n\) generators \(z_1,\dots, z_n\) which acts on \(\ell^2({\mathcal F}_n)\) by \(\Lambda(w)\xi_u= \xi_{wu}\) for \(v\), \(w\) in \({\mathcal F}_n\).
The authors prove that automorphisms of \({\mathcal L}_n\) are automatically norm and WOT continuous and there is a natural homomorphism from \(\operatorname{Aut}({\mathcal L}_n)\) onto \(\operatorname{Aut}(B_n)\), the group of conformal automorhisms of the ball of \(C^n\), the kernel of this map is the ideal of automorphisms which are trivial modulo the WOT-closed commutator ideal.
The following theorem is interesting:
Theorem. For each \(\theta\) in \(\operatorname{Aut}({\mathcal L}_n)\), the dual map \(\tau_\theta\) on \(\text{Rep}_1({\mathcal L}_N)\) given by \(\tau_\theta(\varphi)= \varphi\circ\theta^{-1}\) maps the open ball \(B_n\) conformally onto itself. This determines a homomorphism of \(\operatorname{Aut}({\mathcal L}_n)\) into the group \(\operatorname{Aut}(B_n)\) of conformal automorphisms.
(We identify \(B_n\) with \(\text{Rep}_1({\mathcal L}_n)\) by associating \(\lambda\) in \(B\) with the multiplicative linear functional \(\varphi_\lambda\) in \(\text{Rep}_1({\mathcal L}_n)\)).