Let $H$ be an infinite dimensional seperable Hilbert space. Is there an Irreducible involutive sub algebra $D$ of $B(H)$ with the following properties?:
1)For every open set $U\subset H$ and every Frechet differential map $f:U \to H$ with $Df(x)\in D,\; \forall x\in U$, the mapping $f$ is automatically $C^{\infty}$
2)For every open set $U\subset H$, a uniform limit $f$ of a sequence of Frechet differential maps $f_n:U \to H$ with $Df_n(x)\in D$ is Frechet differentiable which satisfies $Df(x)\in D,\;\forall x\in U$.
The motivation comes from the concepet of holomorphic functions when we put (the real analogy) $ H=\mathbb{R}^2$ and $D=\left \{\begin{pmatrix} a&b\\ -b&a\end{pmatrix} \mid a,b\in \mathbb{R} \right \}$