I'm reading the paper "Tensor products for monotone complete C*-algebras, I" by Hamana. Consider the following fragment in this paper:
Why exactly, in the proof of lemma 3.5, is the sum $$\sum_{\alpha, \beta}\phi(x_{\alpha,\beta})\otimes f_{\alpha, \beta}$$ convergent in the strong topology? All the author shows is that the partial sums are uniformly bounded. Surely this is not enough to conclude strong convergence of the sum?
Thanks in advance for any help!
For a series of the form $$\tag1 \sum_\alpha \eta_\alpha\otimes\delta_\alpha $$ to exist in $H\otimes K$, we need the net of partial sums to be Cauchy. If $G\subset F$ are finite sets, $$ \Big\|\sum_F \eta_\alpha\otimes\delta_\alpha-\sum_G\eta_\alpha\otimes\delta_\alpha\Big\|^2 =\Big\|\sum_{F\setminus G}\eta_\alpha\otimes\delta_\alpha\Big\|^2 =\sum_{F\setminus G}\|\eta_\alpha\|^2. $$ So convergence of the series $\sum_\alpha\|\eta_\alpha\|^2$ is a sufficient (and necessary) condition for the existence of $(1)$ in $H\otimes K$.
The hypothesis is that $x=\sum_{\alpha,\beta}x_{\alpha,\beta}\otimes f_{\alpha,\beta}$ exists as a sot-limit. Evaluating at $\eta\otimes \delta_\gamma$, $\def\abajo{\\[0.3cm]}$ \begin{align} \|x\|^2\,\|\eta\|^2 &\geq\|x(\eta\otimes\delta_\gamma)\|^2 = \Big\|\sum_{\alpha}x_{\alpha,\gamma}\eta\otimes\delta_\alpha\Big\|^2\abajo &=\sum_{\alpha,\beta}\langle x_{\alpha,\gamma}\eta,x_{\beta,\gamma}\eta\rangle\,\langle\delta_\alpha,\delta_\beta\rangle\abajo &=\sum_\alpha \langle x_{\alpha,\gamma}^*x_{\alpha,\gamma}\eta,\eta\rangle. \end{align} This shows that the partial sums of $\sum_\alpha x_{\alpha,\gamma}^*x_{\alpha,\gamma}$ form an increasing bounded net of positive operators, hence sot-convergent, and $$\sum_\alpha x_{\alpha,\gamma}^*x_{\alpha,\gamma}\leq\|x\|^2\,1.$$ Now, for any finite index set $F$, and using that $\phi$ is $2$-positive (for Kadison's inequality), \begin{align} \sum_{\alpha\in F}\|\phi(x_{\alpha,\gamma})\eta\|^2 &=\sum_{\alpha\in F}\langle\phi(x_{\alpha,\gamma}^*)\phi(x_{\alpha,\gamma})\eta,\eta\rangle\abajo &\leq \sum_{\alpha\in F}\langle\phi(x_{\alpha,\gamma}^*x_{\alpha,\gamma})\eta,\eta\rangle\abajo &=\langle\phi(\sum_{\alpha\in F} x_{\alpha,\gamma}^*x_{\alpha,\gamma})\eta,\eta\rangle\abajo &\leq \langle \phi(\|x\|^2\,1)\eta,\eta\rangle\abajo &=\|x\|^2\,\|\eta\|^2. \end{align} Hence $\sum_\alpha\|\phi(x_{\alpha,\gamma})\eta\|^2<\infty$ and $\sum_\alpha\phi(x_{\alpha,\gamma})\eta\otimes\delta_\alpha$ converges in $H\otimes K$.
As noted by @QuantumSpace, if $\phi$ is not unital the computations still go through, just a $\|\phi\|$ appears when one does Kadison's inequality. And they still go through for completely bounded $\phi$, as it will be linear combination of completely positive maps.