Suppose $V$ is a linear subspace of a finite dimensional $C^*$-algebra $A$. (Feel free to assume $A$ is a multi-matrix algebra over $\mathbb C$.)
I would like to find $x \in V$ such that $\mathbb C [x] \subset V$ (that is, the algebra generated by $x$ stays inside $V$).
Are there good numerical methods for locating potential solutions?
Of course, for most subspaces $V$ there are no such $x$ except $0$. In the instances of this problem I'm facing I expect there to be solutions, and I'm more interesting in finding some than showing there aren't any. I'm happy to only look for self-adjoint solutions.
In my target applications, $V$ is large enough that just looking at the polynomial equations expressing $x^n \in V$ for some small values of $n$ is way beyond the range of Groebner bases or numeric algebraic geometry. I think I'll need something that uses more of the structure of the problem, and I'm hoping it's something people have looked at before.