Let $H$ be a seperable complex Hilbert space. What is the closure of the set of all self adjoint operators in $B(H)$ whose spectrum is a subset of the rational number $\mathbb{Q}$.
Apart from finite dimensional $C^*$ algebras what are some other examples of algebra with the following property:
"Every self adjoint element can be approximated with a self adjoint $a$ with $\delta(a)\subset \mathbb{Q} $"
A unital C*-algebra has real rank zero if the self-adjoint elements with finite spectrum are norm-dense in the set of all self-adjoints. If a self-adjoint element has finite spectrum, then it is of the form $a=\sum_{j=1}^n \lambda_j p_j$ for $\lambda_j\in\mathbb{R}$ and pairwise orthogonal projections $p_j$ that sum to the unit. Choosing rational numbers $\lambda_j'$ close to $\lambda_j$, we see that $a$ is norm-close to $\sum_{j=1}^n \lambda_j' p_j$, which has finite spectrum contained in $\mathbb{Q}$.
Thus, in every unital C*-algebra of real rank zero, every self-adjoint element can be approximated by self-adjoint elements with finite spectrum contained in $\mathbb{Q}$.
Every von Neumann algebra (in particular, $B(H)$) has real rank zero.
