On automorphisms of the Toeplitz algebra (Q2702994)

The Toeplitz algebra \(\mathcal T(L^\infty)\) is the C*-algebra generated by Toeplitz operators \(T_\varphi, \varphi \in L^\infty(\mathbb S^1)\), where by definition, \(T_\varphi = PM_\varphi |_{H^2}\), \(P\) is the projection from \(L^2(\mathbb S^1)\) onto the Hardy subspace \(H^2(\mathbb S^1) \subset L^2(\mathbb S^1)\), \(M_\varphi\) is the multiplication by \(\varphi\) operator. An absolutely continuous homeomorphism \(\sigma\) of the circle \(\mathbb S^1\) is a bi-Lipschitz homeomorphism iff its derivative with respect to arc length is essentially bounded and essentially bounded away from zero. NEWLINENEWLINENEWLINEThe main result of the paper under reviewing is ``to complete the proof'' of the result (Theorem 1.1) that if \(\sigma\) is an orientation preserving, bi-Lipchitz homeomorphism of the circle \(\mathbb S^1\), then there is a *-automorphism \(\alpha_\sigma\) of \(\mathcal T(L^\infty)\) such that \(\alpha_\sigma(T_\varphi)\equiv T_{\varphi \circ \sigma}\) (\text{modulo compact operators}), for all \(\varphi \in L^\infty(\mathbb S^1)\), if and only if its derivative \(\sigma'\) with respect to arc length is in the so called VMO class [\textit{D. Sarason}, Trans. Am. Math. Soc. 207, 391-405 (1975; Zbl 0319.42006)] of functions with vanishing mean oscillation. Previously, in [\textit{P. Muhly} and \textit{J. Xia}, Am. J. Math. 117, 1157-1201 (1995; Zbl 0854.42012)] the authors proved the ``sufficient part'' of this criterion.