Aroujo stated the following four open questions at the end of his paper, page $ 518$ and $ 519.$

Question $ 1:$ Assume that there exists a biseparating map $ T:A^n(\Omega:E)\to A^m(\Omega’,F)$ which is not linear. Can we deduce that the support map $ h:\Omega’\to\Omega$ is a diffeomorphism of class $ n?$

Problem $ 2:$ Suppose that $ C^\infty(\Omega,E)$ is the space of $ E$ -valued functions which are of class $ C^\infty$ in $ \Omega,$ and that $ C^\infty(\Omega’,F)$ is defined in a similar way. Describe the linear biseparating maps from $ C^\infty(\Omega,E)$ onto $ C^\infty(\Omega’,F).$ Must such map be continuous?

Problem $ 3:$ Let $ \Omega$ and $ \Omega’$ be unbounded open subsets of $ \mathbb{R}^p$ and $ \mathbb{R}^q$ respectively. Describe the linear biseparating maps from $ C^n_*(\Omega,E)$ onto $ C^m_*(\Omega’,F).$

Problem $ 4:$ Determine all subspaces $ A(\Omega,E)\subseteq A^n(\Omega,E)$ and $ B(\Omega’,F)\subseteq A^m(\Omega’,F)$ such that the existence of a linear biseparating map from $ A(\Omega,E)$ onto $ B(\Omega’,F)$ implies that $ E$ and $ F$ are isomorphic as Banach spaces.

What is the status of these questions? Any progress on each of them? If yes, any reference is appreciated.

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Definitions: Let $ \Omega\subseteq \mathbb{R}^p$ be an open subset and $ E$ be a Banach space. Denote $ C^n(\Omega,E)$ the space of $ E$ -valued functions $ f$ on $ \Omega$ that are of class $ C^n.$ $ T:C^n(\Omega,E)\to C^m(\Omega’,F)$ is called biseparating map if $ T$ is bijective and $ $ \|Tf\| \|Tg\|=0 \text{ if and only if }\|f\|\|\|g\|=0$ $