Adjoint of series of partial isometries

Question

let $\{v_i\}_i$ be a family of pairwise orthogonal partial isometries in a von Neumann algebra $M$.

1. Why does series $\sum_i v_i$ converge strongly to a partial isometry in $M$?
2. Why is the adjoint $(\sum_i v_i)^*$ given by $\sum_i v_i^*$?

Regarding the latter: I know that involution is not strongly continuous, otherwise that would be even clear to me.

I'm suspicious because I cannot use something like the order-completeness of the hermitian elements in von Neumann-algebras since partial isometries need not be hermitian I think...

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So far: I think one can just employ the definition of an adjoint operator and come up with the idea that the adjoint of the series is as indicated above : $<(\sum_i v_i)^* x,y>=\dots=<\sum_i (v_i^*x),y>$ is that correct?

If so : does that already tell me that $(\sum_i v_i)^*$ is the strong limit of the series (thought of as a net of partial sums) $\sum_i (v_i^*x)$?

Answer 1

The assumption is that both the domains and the codomains are pairwise orthogonal. In that case, you basically have a direct sum.

For a proof, denote $p_j=v_j^*v_j$, $q_j=v_jv_j^*$. It is standard that $\sum p_j=p$ and $\sum q_j=q$ converge strongly. Given $x\in H$, $F\subset G$ finite sets of indices, \begin{align} \left\|\sum_{j\in G}v_jx-\sum_{j\in F}v_jx\right\|^2 &=\left\langle\sum_{j\in G}v_jx,\sum_{j\in G}v_jx \right\rangle +\left\langle\sum_{j\in F}v_jx,\sum_{j\in F}v_jx \right\rangle -2\operatorname{Re}\left\langle\sum_{j\in G}v_jx,\sum_{j\in F}v_jx \right\rangle\\ \ \\ &=\sum_{j\in G}\langle p_jx,x\rangle+\sum_{j\in F}\langle p_jx,x\rangle-2\operatorname{Re}\sum_{j\in G, k\in F}\langle v_k^*v_jx,x\rangle\\ \ \\ &=\sum_{j\in G}\langle p_jx,x\rangle+\sum_{j\in F}\langle p_jx,x\rangle-2\sum_{j\in F}\langle p_jx,x\rangle\\ \ \\ &=\sum_{j\in G}\langle p_jx,x\rangle-\sum_{j\in F}\langle p_jx,x\rangle\\ \ \\ &=\sum_{j\in G\setminus F}\langle p_jx,x\rangle =\left\langle \sum_{j\in G\setminus F} p_j\,x,x\right\rangle. \end{align} It follows that the net of partial sums is Cauchy, and so convergent. Then element $\sum_jv_j$ is indeed, in $B(H)$, since \begin{align} \Big\|\sum_jv_jx\Big\|^2&=\sum_{k,j}\langle v_jx,v_kx\rangle=\sum_{j}\langle v_jx,v_jx\rangle=\sum_j\langle v_j^*v_jx,x\rangle\\[0.3cm] &=\sum_j\langle p_jx,x\rangle=\sum_j\langle p_jx,p_jx\rangle=\sum_j\|p_jx\|^2\leq\|x\|^2. \end{align}

For the adjoint, \begin{align} \left\langle \left( \sum_j v_j\right)^*x,x\right\rangle &=\left\langle x,\left( \sum_j v_j\right)x\right\rangle =\sum_j\langle x,v_jx\rangle =\sum_j\langle v_j^*x,x\rangle=\left\langle\sum_j v_j^*\,x,x\right\rangle. \end{align} The very last equality requires that the $q_j$ are pairwise orthogonal to guarantee the convergence of the series.