Representation of algebras as bounded nilpotent operators

Question

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 vector subspace of nilpotent bounded operators, i.e., such that $T^2=0$.

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 $\operatorname{Nil}(\mathcal{H})\hookrightarrow\operatorname{B}(\mathcal{H})$?

In other words: Under which conditions there exists an algebra homomorphism $\rho:A\rightarrow \operatorname{B}(\mathcal{H})$ whose image is contained in $\operatorname{Nil}(\mathcal{H})$?

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.

Answer 1

The problem is still ill-posed, because the set of nilpotent bounded operators is not a vector space. (The sum of two nilpotents need not be nilpotent.) Let's interpret the question as: which algebras admit faithful representations as algebras of nilpotent bounded operators?

Note that "nilpotent" usually means $T^n = 0$ for some $n$, not $T^2 = 0$. But whatever definition you take, it will automatically be shared by $A$ and any faithful representation of $A$. So the question is just: which nilpotent algebras can be faithfully represented as algebras of bounded operators?

In the finite dimensional case, a theorem of Cayley tells us that every algebra can be faithfully represented as an algebra of matrices. So in this case every nilpotent algebra has a faithful matrix representation. I think the same is true of any algebra whose dimension is countable. Sketch of proof: assume $A$ is unital and let $T_1, T_2, \ldots$ be a vector space basis for $A$. We regard $A$ as acting on itself by left multiplication and need to find an inner product with respect to which every left multiplication operator is bounded. We can do this by taking the $T_i$ to be mutually orthogonal, and inductively taking the Hilbert space norm of $T_i$ to be sufficiently large so as to ensure that the compression of $M_{T_j}$ (the left multiplication operator), for any $j < i$, to ${\rm span}(T_1, \ldots, T_i)$ has norm at most $(2-1/i)r_j$ where $r_j$ is the norm of its compression to ${\rm span}(T_1, \ldots, T_j)$.