What we will be concerned with is, first, the question of the condition about $mathcal{L}$ that gives Alg$mathcal{L}$ a von Neumann algebra, that is, the question of the condition about $mathcal{L}$ that will give Alg$mathcal{L}$ a self-adjoint algebra. Secondly, if Alg$mathcal{L}$ is a von Neumann algebra, we want to find out what type it is.