Definition of von Neumann subalgebra

Question

Let $\mathcal M\subseteq B(\mathcal H)$ be a von Neumann algebra. Then, what should be natural definition of a von Neumann subalgebra?

Answer 1

I'd say a von Neumann subalgebra of a von Neumann algebra $\mathcal{M}$ on a Hilbert space $\mathcal{H}$ should at least be a subset $\mathcal{N}$ of $\mathcal{M}$ that is itself a von Neumann algebra—or in other words, a weakly closed $C^*$-subalgebra of $\mathcal{M}$. But one might perhaps require in addition that $\mathcal{N}$ contains the unit of $\mathcal{M}$ (depending on the context.)

This definition of von Neumann subalgebra seems (but doesn't) depend on the choice of the Hilbert space $\mathcal{H}$. Indeed, one can show (using Lemma 1 of this paper of Kadison's) that a $C^*$-subalgebra $\mathcal{N}$ of a von Neuman algebra $\mathcal{M}$ is a von Neumann subalgebra of $\mathcal{M}$ if and only if for every bounded directed set $D$ of self-adjoint elements of $\mathcal{M}$ the supremum $\bigvee D$ in $\mathcal{M}$ of $D$ is in $\mathcal{N}$. Although the supremum $\bigvee D$ is just the strong (or weak) limit of the net $(d)_{d\in D}$, the supremum $\bigvee D$ does not depend not on $\mathcal{H}$ (being defined in terms of the order on $\mathcal{M}$.)