Automorphism group on the Toeplitz algebras (Q1919022)

Let \(S^{2n-1}\) be the unit sphere of \(\mathbb{C}^n\), \({\mathcal T}(C(S^{2n-1}))\) the \(C^*\)-algebra generated by the Toeplitz operators with symbols in \(C(S^{2n-1})\) and \(\Aut({\mathcal T}(C(S^{2n-1})))\) the \(*\)-automorphism group of \({\mathcal T}(C(S^{2n-1}))\). In the case \(n=1\), \({\mathcal T}(C(S^1))\) is the \(C^*\)-algebra generated by the shift \(T_z\), and it is well known that the quotient of \(\Aut{\mathcal T}(C(S^1))\) and its normal subgroup \[ \{W\in U(H^2)\mid [W,T_z]\in{\mathcal K}(H^2)\} \] is isomorphic to \(\text{Homeo}_+ (S^1)\), where \(U(H^2)\) is the unitary group on \(H^2\), \({\mathcal K}(H^2)\) the ideal of compact operators on \(H^2\) and \(\text{Homeo}_+(S^1)\) is the group of orientation preserving homeomorphisms on \(S^1\) [\textit{L. G. Brown, R. G. Douglas} and \textit{P. A. Fillmore}, Lect. Notes Math. 345, 58-128 (1973; Zbl 0277.46053)]. In the present note, the authors generalize the above result to the cases \({\mathcal T}(C(S^{2n-1}))\) for all \(n\geq 1\).