Let $A$ be a real/complex algebra (just a real/complex vector space with a multiplication; none PI's are required). Let $\mathcal{H}$ be a real/complex Hilbert space. Let $\operatorname{B}(\mathcal{H})$ be the algebra of bounded endomorphisms of $\mathcal{H}$ and let $\operatorname{Nil}(\mathcal{H})\subset\operatorname{B}(\mathcal{H})$ be the subalgebra of mutually commuting nilpotent bounded operators. This means that $T^2=0$ and $[T,T']=0$ for every $T,T'\in \operatorname{Nil}(\mathcal{H})$.

My question is:

There are known conditions on $A$ and on $\mathcal{H}$ to ensure the existence of representations $\rho:A\rightarrow \operatorname{B}(\mathcal{H})$ which factor trough the inclusion $\operatorname{Nil}(\mathcal{H})\hookrightarrow\operatorname{B}(\mathcal{H})$?

Thank you for any help.

P.S: I'm specially interested when the Hilbert space is the Sobolev space of some nice bounded open set $U\subset \mathbb{R}^n$ and related structures.