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The natural functions in this context are Lipschitz; e.g. functions $d_x(y)=d(x,y)$ determine an isometric embedding of a metric space $X$ to $L_\infty(X)$. For a noncommutative analogue see G.Kuperberg, N.Weaver, "A von Neumann algebra approach to quantum metrics", http://front.math.ucdavis.edu/1005.0353https://arxiv.org/abs/1005.0353

The natural functions in this context are Lipschitz; e.g. functions $d_x(y)=d(x,y)$ determine an isometric embedding of a metric space $X$ to $L_\infty(X)$. For a noncommutative analogue see G.Kuperberg, N.Weaver, "A von Neumann algebra approach to quantum metrics", http://front.math.ucdavis.edu/1005.0353

The natural functions in this context are Lipschitz; e.g. functions $d_x(y)=d(x,y)$ determine an isometric embedding of a metric space $X$ to $L_\infty(X)$. For a noncommutative analogue see G.Kuperberg, N.Weaver, "A von Neumann algebra approach to quantum metrics", https://arxiv.org/abs/1005.0353

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Misha
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The natural functions in this context are Lipschitz; e.g. functions $d_x(y)=d(x,y)$ determine an isometric embedding of a metric space $X$ to $L_\infty(X)$. For a noncommutative analogue see G.Kuperberg, N.Weaver, "A von Neumann algebra approach to quantum metrics", http://front.math.ucdavis.edu/1005.0353