bibkey: peterson2013monotonenet authors: Jesse Peterson year: 2013 title: Notes on von Neumann algebras doi: null url: https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf claim: An increasing norm-bounded net of positive operators on a complex Hilbert space converges strongly; its strong limit is its operator-order least upper bound. strata_touched:

• D5/S3/Quantum/Measurements/DirectedPositiveNet license: citation-only triage: anchor

Monotone convergence of positive operator nets

Peterson, Notes on von Neumann algebras, Lemma 2.7.1, states strong convergence of an increasing norm-bounded net of positive operators on a Hilbert space. The formulation below starts from an existing operator-order least upper bound. The square-root estimate is the estimate used in Peterson’s proof; construction from vector Cauchy limits and identification of the least upper bound are written out here. No countable cofinal subset is required.

Theorem 1 (strong convergence to the operator-order least upper bound)

Let $H$ be any complete complex inner-product space, including the zero space, and let $J$ be a nonempty directed preorder. Let $X:J\to B(H)$ be a monotone net of positive operators. Suppose $U\in B(H)$ is the actual least upper bound of $\{X_j:j\in J\}$ for the positive-operator order. Then

$$\forall j\in J,\Vert X_j\Vert \le \Vert U\Vert ,\forall v\in H,X_{j}v⟶Uv(j\to \infty ).$$

Proof. Positivity and $X_j\le U$ imply the norm bound. If $D\ge 0$, set $R=D^{1/2}$. Self-adjointness of $R$, $R^2=D$, and the C-star identity give

$$\Vert Dv{\Vert }^2=\Vert R(Rv){\Vert }^2\le \Vert R{\Vert }^2\Vert Rv{\Vert }^2=\Vert D\Vert \mathrm{Re}⟨Dv,v⟩.$$

For fixed $v$, the real scalar net $q_j(v)=\mathrm{Re}⟨X_{j}v,v⟩$ is monotone and bounded above by $\mathrm{Re}⟨Uv,v⟩$. Thus it converges to its real supremum $Q(v)$. This does not yet identify $Q(v)$ with the quadratic form of $U$.

Set $C=\Vert U\Vert +1>0$. For $i\le k$, positivity yields $0\le X_k-X_i\le U$ and hence

$$\Vert X_{k}v-X_{i}v{\Vert }^2\le C(q_k(v)-q_i(v))\le C(Q(v)-q_i(v)).$$

Given $\epsilon >0$, choose $N$ such that $Q(v)-q_i(v)<{\epsilon }^2/(16C)$ whenever $i\ge N$. For $i,j\ge N$, directedness gives $k\ge i,j$. The displayed estimate and the triangle inequality give $\Vert X_{i}v-X_{j}v\Vert <\epsilon$. Thus the vector net is Cauchy and has a limit $s(v)$. Uniqueness of vector limits proves additivity and complex linearity of $s$. Passing the operator-norm bound to the limit gives $\Vert s(v)\Vert \le \Vert U\Vert \Vert v\Vert$, so $s$ defines a bounded linear operator $S$.

The cone of positive operators is closed under pointwise strong limits: for every $v$, the nonnegative complex numbers $⟨F_{j}v,v⟩$ converge to $⟨Fv,v⟩$, which is real and nonnegative. For fixed $i$, the net $X_j-X_i$ is eventually positive, so $S-X_i$ is positive. If $V$ is any upper bound of every $X_j$, then $V-X_j$ is positive for every $j$, so $V-S$ is positive. Consequently $S$ is a least upper bound. Uniqueness of least upper bounds gives $S=U$, proving the claimed strong convergence. The positive constant $C$ also covers $U=0$; none of the argument excludes $H=0$.

Verified locator

https://math.vanderbilt.edu/peters10/teaching/spring2013/vonNeumannAlgebras.pdf

Jesse Peterson, Notes on von Neumann algebras, April 5, 2013, Lemma 2.7.1, page 32. The original PDF states strong convergence for an increasing norm-bounded net of positive operators. The operator-order least-upper-bound identification is derived in Theorem 1 above.