number of generators of MASA

Question

Let $\mathcal{H}$ be an infinite-dimensional Hilbert Space. Do the maximal abelian self-adjoint subalgebras of $\mathcal{B}(\mathcal{H})$ always have infinitely many generators as an algebra ? (The question may also be interpreted in the sense of taking closure of the generated algebra.)

Answer 1

The two questions have different answers:

If we consider only the algebra generated by the generators and do not take any closure:

As Jonas Meyer pointed out in his comment, and as can be seen in this question about the cardinality of a Hamel basis, a complete infinite dimensional space has uncountable Hamel dimension and would require infinitely many generators.

If we can take the weak closure:

An abelian von Neumann algebra on a separable Hilbert space is generated by a single self-adjoint operator.

This is Proposition III.1.21 in Takesaki's Theory of Operator Algebras, Volume 1. That is, every such algebra is the weak-closure of the algebra generated by a single self-adjoint element.