Geodesic
Isomorphisms of Royden Type Algebras Over $\mathbb{S}^1$
Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 719-729.

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Let $\mathbb{S}^{1}$ and $\mathbb{D}$ be the unit circle and the unit disc in the plane and let us denote by $\mathcal{A}(\mathbb{S}^{1})$ the algebra of the complex-valued continuous functions on $\mathbb{S}^{1}$ which are traces of functions in the Sobolev class $W^{1,2}(\mathbb{D})$. On $\mathcal{A}(\mathbb{S}^{1})$ we define the following norm \begin{equation*} \|u\| = \|u\|_{L^{\infty}(\mathbb{S}^{1})} + \left(\iint _{\mathbb{D}} |\nabla \tilde{u}|^{2} \right)^{1/2} \end{equation*} where is the harmonic extension of $u$ to $\mathbb{D}$. We prove that every isomorphism of the functional algebra $\mathcal{A}(\mathbb{S}^{1})$ is a quasitsymmetric change of variables on $\mathbb{S}^{1}$. 
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Radice, Teresa; Saksman, Eero; Zecca, Gabriella. Isomorphisms of Royden Type Algebras Over $\mathbb{S}^1$. Bollettino della Unione matematica italiana, Série 9, Tome 2 (2009) no. 3, pp. 719-729. http://geodesic.mathdoc.fr/item/BUMI_2009_9_2_3_a11/

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