Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and self-adjoint, and is maximal with respect to these conditions.
MASAs are quite standard materials in operator theory, but I wonder whether we have studied MAAs, that is, we drop the condition that the algebra has to be closed under involution.
Just like for MASAs, MAA exists if we assume Zorn's lemma. But MASAs can be constructed by hand (for instance, $\mathcal{H}=\mathcal{L}^2(X,\mu)$, a MASA is the multiplication algebra). I am not sure whether we have such kind of explicit examples of MAA.
SO, do we have a nice example of MAA (on $\mathcal{L}^2$ maybe)? Do we know something about MAAs?
Thanks!