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We prove that the space $L^{\\infty} (\\Omega,\\mu)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $\\Delta: S(L^{\\infty} (\\Omega,\\mu))\\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\\infty} (\\Omega,\\mu)\\to X$. 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Peralta, Mar\\'ia Cueto-Avellaneda","submitted_at":"2018-03-01T20:05:12Z","abstract_excerpt":"Let $(\\Omega,\\mu)$ be a $\\sigma$-finite measure space. Given a Banach space $X$, let the symbol $S(X)$ stand for the unit sphere of $X$. We prove that the space $L^{\\infty} (\\Omega,\\mu)$ of all complex-valued measurable essentially bounded functions equipped with the essential supremum norm, satisfies the Mazur-Ulam property, that is, if $X$ is any complex Banach space, every surjective isometry $\\Delta: S(L^{\\infty} (\\Omega,\\mu))\\to S(X)$ admits an extension to a surjective real linear isometry $T: L^{\\infty} (\\Omega,\\mu)\\to X$. 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